# Polygon: Wikis

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# Encyclopedia

An assortment of polygons
.In geometry a polygon (pronounced /ˈpɒlɪɡɒn/) is traditionally a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain).^ A polygon is a closed figure composed of three or more line segments that intersect at their endpoints.
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ The symbol , which includes the arrow heads at both ends indicates the whole line where , which does not have the arrow heads, indicates a line segment, which is finite in length (only the part of the line from A to B).
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ The 'polyline' element defines a set of connected straight line segments.
• Basic Shapes - SVG 1.1 - 20030114 12 January 2010 2:53 UTC www.w3.org [Source type: Reference]

.These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners.^ Each line segment is called the side .
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ What could be inside a two-sided polygon?

^ In all cases, the polygon is specified as an array of X,Y coordinates of the corner points.

The interior of the polygon is sometimes called its body. .A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.^ For this method, num points sets the minimum number of points that a generated polygon will have (if the polygon were to have fewer, it will not be generated at all).
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

.The word "polygon" derives from the Greek πολύς ("many") and γωνία (gōnia), meaning "knee" or "angle". Today a polygon is more usually understood in terms of sides.^ Polygon is from the Greek roots poli (many) and gonus (knees) and, interprets literally as many angled.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ Question 254550 : how many sides does a polygon have if thhe sum of its interior angles is 540?
• Questions on Geometry: Polygons answered by real tutors! 12 January 2010 2:53 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ The word still retains that meaning today in phonetics for an unvoiced consonant (as opposed to a voiced consonant, a sonant).
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

.Usually two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge.^ Alternate interior angles are = *Straight angle = 180 .
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ The symbol , which includes the arrow heads at both ends indicates the whole line where , which does not have the arrow heads, indicates a line segment, which is finite in length (only the part of the line from A to B).
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ A straight angle is one that measures exactly 180 degrees: .
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

The basic geometrical notion has been adapted in various ways to suit particular purposes. .For example in the computer graphics (image generation) field, the term polygon has taken on a slightly altered meaning, more related to the way the shape is stored and manipulated within the computer.^ There are several toggles in the EditPolygon dialog which control how polygon graphics are displayed in the image window.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ Because of the nature of the search, it is not necessary for the seed point to lie within the polygon which is generated.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ Auto Clicking in the image window causes the point selected to be used as the starting point for automatically generating a polygon.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

## Classification

### Number of sides

.Polygons are primarily classified by the number of sides, see naming polygons below.^ The applet shows a regular polygon where the user can drag the vertices to reshape it and alter the number of sides.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The applet shows a polygon where the user can drag any vertex and change the number of sides.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The user can also alter the number of sides from 3 to 99, the title changing to reflect it's name up to 12 sides.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

### Convexity

Polygons may be characterised by their degree of convexity:
.
• Convex: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice.
• Non-convex: a line may be found which meets its boundary more than twice.
• Simple: the boundary of the polygon does not cross itself.^ The polygon that is generated can be larger or smaller than the boundary found using any of the methods.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ Rod may chime in here, he knows much more detail about the program than I do.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

^ If this transform represents a non-uniform scale or more general transform then the determinant is not likely to represent a value useful for any purpose other than determining if inverse transforms are possible.
• AffineTransform (Java 2 Platform SE v1.4.2) 11 September 2009 9:28 UTC java.sun.com [Source type: Reference]

.All convex polygons are simple.
• Concave: Non-convex and simple.
• Star-shaped: the whole interior is visible from a single point, without crossing any edge.^ When on, all polygons in the currently displayed z section and time point are displayed, regardless of the wave.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ Smoothes the currently selected polygon by operating on all the jagged edges at once.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ When EditPolygon is in the autopolygon mode, rather than manually selecting all the vertices of a polygon, a seed point is selected and that is used to generate the polygon (or polygons if a 3D search is enabled).
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

The polygon must be simple, and may be convex or concave.
• Self-intersecting: the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used. The term complex is sometimes used in contrast to simple, but this risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions.
• Star polygon: a polygon which self-intersects in a regular way.

### Symmetry

• Equiangular: all its corner angles are equal.
• Cyclic: all corners lie on a single circle.
• Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. .The polygon is also cyclic and equiangular.
• Equilateral: all edges are of the same length.^ Smoothes the currently selected polygon by operating on all the jagged edges at once.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ It uses the same criteria as change from center to locate a point on the polygon edge to the right of the seed point.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ All of these programs can look at the same set of polygons so that they can be used together.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

(A polygon with 5 or more sides can be equilateral without being convex.) (Williams 1979, pp. .31-32)
• Isotoxal or edge-transitive: all sides lie within the same symmetry orbit.^ This method computes a binary OR of the appropriate mask values indicating, for each side of this Rectangle , whether or not the specified coordinates are on the same side of the edge as the rest of this Rectangle .
• Rectangle (Java 2 Platform SE v1.4.2) 11 September 2009 9:28 UTC java.sun.com [Source type: FILTERED WITH BAYES]

The polygon is also equilateral.
• Regular. A polygon is regular if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon.

### Miscellaneous

• Rectilinear: a polygon whose sides meet at right angles, i.e., all its interior angles are 90 or 270 degrees.
• Monotone with respect to a given line L, if every line orthogonal to L intersects the polygon not more than twice.

## Properties

We will assume Euclidean geometry throughout.

### Angles

.Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides.^ The applet shows a regular polygon where the user can drag the vertices to reshape it and alter the number of sides.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The applet shows a regular polygon where the user can alter the number of sides and resize it by dragging any vertex.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The user can drag any vertex, change the number of sides in the range 3..99, and make it regular or irregular.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

Each corner has several angles. The two most important ones are:
.
• Interior angle – The sum of the interior angles of a simple n-gon is (n − 2)π radians or (n − 2)180 degrees.^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The web page lists the properties of a dodecagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ (Complete Item Description) Abstract: An interactive applet and associated web page that demonstrate the concept of a convex polygon - one where all interior angle are less than 180 degrees.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

.This is because any simple n-gon can be considered to be made up of (n − 2) triangles, each of which has an angle sum of π radians or 180 degrees.^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The web page lists the properties of a dodecagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The web page lists the properties of an octagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

.The measure of any interior angle of a convex regular n-gon is (n − 2)π/n radians or (n − 2)180/n degrees.^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ User can see that the interior and exterior angles are constant in a regular octagon, but vary in an irregular version.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ User can see that the interior and exterior angles are constant in a regular pentagon, but vary in an irregular version.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

.The interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra.
• Exterior angle – Imagine walking around a simple n-gon marked on the floor.^ Interior / exterior angles of a polygon .
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The web page lists the properties of a dodecagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

The amount you "turn" at a corner is the exterior or external angle. .Walking all the way round the polygon, you make one full turn, so the sum of the exterior angles must be 360°.^ Interior / exterior angles of a polygon .
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ That one is great fun and familiarity with it makes you a pro.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

.Moving around an n-gon in general, the sum of the exterior angles (the total amount one "turns" at the vertices) can be any integer multiple d of 360°, e.g.^ At least I can make a new one, move an entity to it, turn it on and off.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

720° for a pentagram and 0° for an angular "eight", where d is the density or starriness of the polygon. See also orbit (dynamics).
.The exterior angle is the supplementary angle to the interior angle.^ Interior / exterior angles of a polygon .
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The web page lists the properties of a dodecagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ User can see that the interior and exterior angles are constant in a regular octagon, but vary in an irregular version.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

.From this the sum of the interior angles can be easily confirmed, even if some interior angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount.^ An object thus beaten becomes blunt, dull, or rounded, as in the application to an obtuse angle, one having more than 90 o but less than 180 o .
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ Some feared John Kerry, others John Edwards, because his personality wears well over time, and others even Bob Graham, because he can carry Florida, more than Dean.
• Polygon, the Dancing Bear 12 January 2010 2:53 UTC potifos.com [Source type: General]

.(Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between −½ and ½ winding.^ The applet shows a polygon where the user can drag any vertex and change the number of sides.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The user can drag any vertex and change the number of sides in the range 3..99.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ The applet shows a regular polygon where the user can alter the number of sides and resize it by dragging any vertex.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

)

### Area and centroid

Nomenclature of a 2D polygon.
.The area of a polygon is the measurement of the 2-dimensional region enclosed by the polygon.^ Commands BOUNDARY Creates a region or a polyline from an enclosed area.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

For a non-self-intersecting (simple) polygon with n vertices, the area and centroid are given by[1]:
$A = \frac{1}{2} \sum_{i = 0}^{n - 1}( x_i y_{i + 1} - x_{i + 1} y_i)\,$
$C_x = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (x_i + x_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i)\,$
$C_y = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (y_i + y_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i)\,$
To close the polygon, the first and last vertices are the same, i.e., xn,yn = x0,y0. The vertices must be ordered clockwise or counterclockwise; if they are ordered clockwise, the area will be negative but correct in absolute value. This is commonly called the Surveyor's Formula.[citation needed]
.The formula was described by Meister[citation needed] in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem.^ Second When the pars minutia [see Minute of an arc needed to be divided into even smaller parts, the 1/60 part of 1/60 of a degree needed a name also.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

The area A of a simple polygon can also be computed if the lengths of the sides, a1,a2, ..., an and the exterior angles, $heta_1, heta_2,\dots, heta_n$ are known. The formula is
\begin{align}A = \frac12 ( a_1[a_2 \sin( heta_1) + a_3 \sin( heta_1 + heta_2) + \cdots + a_{n-1} \sin( heta_1 + heta_2 + \cdots + heta_{n-2})] \ {} + a_2[a_3 \sin( heta_2) + a_4 \sin( heta_2 + heta_3) + \cdots + a_{n-1} \sin( heta_2 + \cdots + heta_{n-2})] \ {} + \cdots + a_{n-2}[a_{n-1} \sin( heta_{n-2})] ) \end{align}
The formula was described by Lopshits in 1963.[2]
.If the polygon can be drawn on an equally-spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.^ When on, all polygons in the currently displayed z section and time point are displayed, regardless of the wave.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ When EditPolygon is in the autopolygon mode, rather than manually selecting all the vertices of a polygon, a seed point is selected and that is used to generate the polygon (or polygons if a 3D search is enabled).
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ The determination of the seed point from the existing polygon is controlled by the using last picked pt toggle.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

.If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon.^ These polygons are used to delineate areas of the data for measurements, for cutting out part of the image, or as building blocks for 3D Objects which can be used for 3D measurement and modeling.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ If you have two overlapping circles and you drop a color in the overlapping area, only that overlapping area is filled, makes it easy to fill overlapping polygons with contrasting colors.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

^ It traces a polygon with a hidden line around the perimeter of the blank area you drop the fill into.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

This is the Bolyai-Gerwien theorem.
For a regular polygon with n sides of length s, the area is given by:
$A = \frac{n}{4} s^2 \cot{\cfrac{\pi}{n}}.$

#### Self-intersecting polygons

The area of a self-intersecting polygon can be defined in two different ways, each of which gives a different answer:
.
• Using the above methods for simple polygons, we discover that particular regions within the polygon may have their area multiplied by a factor which we call the density of the region.^ I had no trouble locating the following description of hatching: " You can hatch an enclosed area or hatch within a specified boundary using HATCH. By default, HATCH creates associative hatches that are updated when the boundary is changed.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

^ Use the method option menu in the autopolygon dialog to select the type of algorithm used to generate a polygon from a seed point.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ This is equivalent to calling concatenate(SH), where SH is an AffineTransform represented by the following matrix: [ 1 shx 0 ] [ shy 1 0 ] [ 0 0 1 ] Parameters: shx - the multiplier by which coordinates are shifted in the direction of the positive X axis as a factor of their Y coordinate shy - the multiplier by which coordinates are shifted in the direction of the positive Y axis as a factor of their X coordinate .
• AffineTransform (Java 2 Platform SE v1.4.2) 11 September 2009 9:28 UTC java.sun.com [Source type: Reference]

.For example the central convex pentagon in the centre of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure.
• Considering the enclosed regions as point sets, we can find the area of the enclosed point set.^ In each of these two instances the lines defining the areas to be filled are a mix of whole line entities and portions of larger line entities.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

^ Invalid Hatch Boundaries When a hatch boundary cannot be determined, it might be because the specified internal point is not within a fully enclosed area.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

^ Another point: improvement in transportation and communications have increasingly centralized our news and our culture at the national level, while eroding local and regional culture or even awareness.
• Polygon, the Dancing Bear 12 January 2010 2:53 UTC potifos.com [Source type: General]

This corresponds to the area of the plane covered by the polygon, or to the area of a simple polygon having the same outline as the self-intersecting one (or, in the case of the cross-quadrilateral, the two simple triangles).

### Degrees of freedom

An n-gon has 2n degrees of freedom, including 2 for position, 1 for rotational orientation, and 1 for over-all size, so 2n − 4 for shape. In the case of a line of symmetry the latter reduces to n − 2.
.Let k ≥ 2. For an nk-gon with k-fold rotational symmetry (Ck), there are 2n − 2 degrees of freedom for the shape.^ There are two major types of self symmetry, rotational (point) and reflective (line).
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

With additional mirror-image symmetry (Dk) there are n − 1 degrees of freedom.

## Generalizations of polygons

In a broad sense, a polygon is an unbounded (without ends) sequence or circuit of alternating segments (sides) and angles (corners). .An ordinary polygon is unbounded because the sequence closes back in itself in a loop or circuit, while an apeirogon (infinite polygon) is unbounded because it goes on for ever so you can never reach any bounding end point.^ The first exercise, where you draw a 2" by 3" rectangle, goes right to my "putting the pencil back on the paper" question.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

.The modern mathematical understanding is to describe such a structural sequence in terms of an "abstract" polygon which is a partially ordered set (poset) of elements.^ Today in mathematics we define a sequence as a group of terms in a row such as 2,4,6,8.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ For example, using the sequence of even terms above, the sequence of partial sums would be 2, 2+4, 2+4+6, etc.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope.
A geometric polygon is understood to be a "realization" of the associated abstract polygon; this involves some "mapping" of elements from the abstract to the geometric. .Such a polygon does not have to lie in a plane, or have straight sides, or enclose an area, and individual elements can overlap or even coincide.^ If you have two overlapping circles and you drop a color in the overlapping area, only that overlapping area is filled, makes it easy to fill overlapping polygons with contrasting colors.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles. .So when we talk about "polygons" we must be careful to explain what kind we are talking about.^ We are only talking about resolution here, bigger characters do not necessarily require more polygons.
• Nintendo May Be Considering SD/HD Hybrid Console 12 January 2010 2:53 UTC www.escapistmagazine.com [Source type: General]

A digon is a closed polygon having two sides and two corners. On the sphere, we can mark two opposing points (like the North and South poles) and join them by half a great circle. .Add another arc of a different great circle and you have a digon.^ Polygons inside polygons To put different fills in concentric polygons (you would have to convert the circles in a donut) select the inside polygon Edit> Trim> Difference and select the outside polygon.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

Tile the sphere with digons and you have a polyhedron called a hosohedron. .Take just one great circle instead, run it all the way round, and add just one "corner" point, and you have a monogon or henagon—although many authorities do not regard this as a proper polygon.^ If the third toggle on the left (the one next to the wave field) is on, the copy is done from all time points z sections in the wave given by the wave field on the left.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

^ That one is great fun and familiarity with it makes you a pro.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

^ If it is set to one , then the polygons you create will only be applied to the currently active wave.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

.Other realizations of these polygons are possible on other surfaces, but in the Euclidean (flat) plane, their bodies cannot be sensibly realized and we think of them as degenerate.^ When any of one of these programs are running and have a set of polygons loaded into an image window, any other one will see those polygons and be able to work with them.
• Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]

The idea of a polygon has been generalized in various ways. Here is a short list of some degenerate cases (or special cases, depending on your point of view):
• Digon. Interior angle of 0° in the Euclidean plane. See remarks above re. on the sphere.
• .
• Interior angle of 180°: In the plane this gives an apeirogon (see below), on the sphere a dihedron
• A skew polygon does not lie in a flat plane, but zigzags in three (or more) dimensions.^ An object thus beaten becomes blunt, dull, or rounded, as in the application to an obtuse angle, one having more than 90 o but less than 180 o .
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

The Petrie polygons of the regular polyhedra are classic examples.
• A spherical polygon is a circuit of sides and corners on the surface of a sphere.
• An apeirogon is an infinite sequence of sides and angles, which is not closed but it has no ends because it extends infinitely.
• A complex polygon is a figure analogous to an ordinary polygon, which exists in the complex Hilbert plane.

## Naming polygons

.The word "polygon" comes from Late Latin polygōnum (a noun), from Greek polygōnon/polugōnon πολύγωνον, noun use of neuter of polygōnos/polugōnos πολύγωνος (the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g.^ The word is a hybrid of Greek and Latin roots.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ The word comes from the Latin rota , for wheel.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ Prime is from the Latin word for first, primus and related to the Greek protos .
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

pentagon, dodecagon. The triangle, quadrilateral or quadrangle, and nonagon are exceptions. For large numbers, mathematicians usually write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.
Some special polygons also have their own names; for example the regular star pentagon is also known as the pentagram.
Polygon names
Name Edges Remarks
henagon (or monogon) 1 In the Euclidean plane, degenerates to a closed curve with a single vertex point on it.
digon 2 In the Euclidean plane, degenerates to a closed curve with two vertex points on it.
triangle (or trigon) 3 The simplest polygon which can exist in the Euclidean plane.
quadrilateral (or quadrangle or tetragon) 4 The simplest polygon which can cross itself.
pentagon 5 The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagram or pentacle.
hexagon 6
heptagon 7 avoid "septagon" = Latin [sept-] + Greek
octagon 8
enneagon (or nonagon) 9
decagon 10
hendecagon 11 avoid "undecagon" = Latin [un-] + Greek
dodecagon 12 avoid "duodecagon" = Latin [duo-] + Greek
tridecagon (or triskaidecagon) 13
pentadecagon (or quindecagon or pentakaidecagon) 15
icosagon 20
No established English name 100 "hectogon" is the Greek name (see hectometre), ."centagon" is a Latin-Greek hybrid; neither is widely attested.^ The word is a hybrid of Greek and Latin roots.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ Etymologically, it means "handed", but is a hybrid, since it combines a Greek stem with the Latin suffix "-al".
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

chiliagon 1000 Pronounced /ˈkɪliəɡɒn/), this polygon has 1000 sides. The measure of each angle in a regular chiliagon is 179.64°.
René Descartes used the chiliagon and myriagon (see below) as examples in his Sixth meditation to demonstrate a distinction which he made between pure intellection and imagination. He cannot imagine all thousand sides [of the chiliagon], as he can for a triangle. However, he clearly understands what a chiliagon is, just as he understands what a triangle is, and he is able to distinguish it from a myriagon. Thus, he claims, the intellect is not dependent on imagination.[3]
myriagon 10,000 See remarks on the chiliagon.
megagon [4] 1,000,000 The internal angle of a regular megagon is 179.99964 degrees.
To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows
Tens and Ones final suffix
-kai- 1 -hena- -gon
20 icosi- 2 -di-
30 triaconta- 3 -tri-
40 tetraconta- 4 -tetra-
50 pentaconta- 5 -penta-
60 hexaconta- 6 -hexa-
70 heptaconta- 7 -hepta-
80 octaconta- 8 -octa-
90 enneaconta- 9 -ennea-
The "kai" is not always used. Opinions differ on exactly when it should, or need not, be used (see also examples above).
Alternatively, the system used for naming the higher alkanes can be used:
Ones Tens final suffix
1 hen- 10 deca- -gon
2 do- 20 -cosa-
3 tri- 30 triaconta-
4 tetra- 40 tetraconta-
5 penta- 50 pentaconta-
6 hexa- 60 hexaconta-
7 hepta- 70 heptaconta-
8 octa- 80 octaconta-
9 ennea- (or nona-) 90 enneaconta- (or nonaconta-)
This has the advantage of being consistent with the system used for 10- thru 19-sided figures.
That is, a 42-sided figure would be named as follows:
Ones Tens final suffix full polygon name
do- tetraconta- -gon dotetracontagon
and a 50-sided figure
Tens and Ones final suffix full polygon name
pentaconta-   -gon pentacontagon
But beyond enneagons and decagons, professional mathematicians generally prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons). Exceptions exist for side numbers that are difficult to express in numerical form.

## History

Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram, a non-convex regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C..[citation needed] Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine.[citation needed]
.In 1952, Shephard[citation needed] generalised the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.^ Mobius work with barycenters present the idea of directed distances, and Argand presentated complex numbers as points or ordered pairs of real numbers.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

## Polygons in nature

The Giant's Causeway, in Ireland
Numerous regular polygons may be seen in nature. In the world of geology, crystals have flat faces, or facets, which are polygons. Quasicrystals can even have regular pentagons as faces. .Another fascinating example of regular polygons occurs when the cooling of lava forms areas of tightly packed hexagonal columns of basalt, which may be seen at the Giant's Causeway in Ireland, or at the Devil's Postpile in California.^ They go on to state that "[b]y the definition of trapezoid here given it will be seen that the parallelogram may be considered a special form of the trapezoid.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

Starfruit, a popular fruit in Southeast Asia
The most famous hexagons in nature are found in the animal kingdom. The wax honeycomb made by bees is an array of hexagons used to store honey and pollen, and as a secure place for the larvae to grow. There also exist animals who themselves take the approximate form of regular polygons, or at least have the same symmetry. For example, sea stars display the symmetry of a pentagon or, less frequently, the heptagon or other polygons. .Other echinoderms, such as sea urchins, sometimes display similar symmetries.^ Reflective symmetry is sometimes called mirror symmetry because one part of the object looks like the reflection of the other half.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

Though echinoderms do not exhibit exact radial symmetry, jellyfish and comb jellies do, usually fourfold or eightfold.
Radial symmetry (and other symmetry) is also widely observed in the plant kingdom, particularly amongst flowers, and (to a lesser extent) seeds and fruit, the most common form of such symmetry being pentagonal. A particularly striking example is the Starfruit, a slightly tangy fruit popular in Southeast Asia, whose cross-section is shaped like a pentagonal star.
.Moving off the earth into space, early mathematicians doing calculations using Newton's law of gravitation discovered that if two bodies (such as the sun and the earth) are orbiting one another, there exist certain points in space, called Lagrangian points, where a smaller body (such as an asteroid or a space station) will remain in a stable orbit.^ Two chiral objects which are reflected images of each other are called enantiomorphic , from the Greek words for opposite, anti'os , and body, morph .
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ She would then cut the monologue short, because there was no point in being excited about your hobbies with such a heavy heart.
• Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]

^ Loculus seems to be a word related to the division of a tomb area into small chambers for different bodies and is related to the diminutive of locus for a point or place, thus "a little place".
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

The sun-earth system has five Lagrangian points. .The two most stable are exactly 60 degrees ahead and behind the earth in its orbit; that is, joining the centre of the sun and the earth and one of these stable Lagrangian points forms an equilateral triangle.^ He wasn't exactly two steps ahead, but with every blow, Tarah could see him get more and more comfortable at being swung at; his cool expression testified to that.
• Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]

^ These are the two tools you'll use the most at the beginning.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

Astronomers have already found asteroids at these points. .It is still debated whether it is practical to keep a space station at the Lagrangian point — although it would never need course corrections, it would have to frequently dodge the asteroids that are already present there.^ She would then cut the monologue short, because there was no point in being excited about your hobbies with such a heavy heart.
• Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]

There are already satellites and space observatories at the less stable Lagrangian points.

## Uses

• Cut up a piece of paper into polygons, and put them back together as a tangram.
• Join many edge-to-edge as a tiling or tessellation.
• Join several edge-to-edge and fold them all up so there are no gaps, to make a three-dimensional polyhedron.
• Use computer-generated polygons to build up a three-dimensional world full of monsters, theme parks, aeroplanes or anything; see Polygons in computer graphics below.

### In computer graphics

A polygon in a computer graphics (image generation) system is a two-dimensional shape that is modelled and stored within its database. A polygon can be coloured, shaded and textured, and its position in the database is defined by the co-ordinates of its vertices (corners).
Naming conventions differ from those of mathematicians:
• A simple polygon does not cross itself.
• a concave polygon is a simple polygon having at least one interior angle greater than 180°.
• A complex polygon does cross itself.
Use of Polygons in Real-time imagery. The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of the processed data to the display system. Although polygons are two dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation so that as the viewing point moves through the scene, it is perceived in 3D.
Morphing. .To avoid artificial effects at polygon boundaries where the planes of contiguous polygons are at different angle, so called "Morphing Algorithms" are used.^ The answer, c, is most often called the difference or result, but in many applied statistical uses it is also called the residue , or residual , that which remains.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

.These blend, soften or smooth the polygon edges so that the scene looks less artificial and more like the real world.^ She wondered if it was less that they were trying to win and more that they simply didn't like her.
• Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]

^ The chill that was less like glacier water and more like icy fire, flickering as a blazing knife deep in his heart.
• Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]

Meshed Polygons. The number of meshed polygons ("meshed" is like a fish net) can be up to twice that of free-standing unmeshed polygons, particularly if the polygons are contiguous. .If a square mesh has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2n squared triangles since there are two triangles in a square.^ There are two major types of self symmetry, rotational (point) and reflective (line).
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

There are (n+1) 2/2n2 vertices per triangle. Where n is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines).
Polygon Count. .Since a polygon can have many sides and need many points to define it, in order to compare one imaging system with another, "polygon count" is generally taken as a triangle.^ Then, if I occupy one of the markers, backsight on another, as in the paragraph above, and lay out a bunch of surveyed points, it seems that I will have to rotate the entire grid around the occupied point in order to make the backsight zero degrees.
• Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]

^ Some geometry textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides, so that a parallelogram is a type of trapezoid.
• Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

When analysing the characteristics of a particular imaging system, the exact definition of polygon count should be obtained as it applies to that system as there is some flexibility in processing which causes comparisons to become non-trivial. Vertex Count. Although using this metric appears to be closer to reality it still must be taken with some salt. Since each vertex can be augmented with other attributes (such as color or normal) the amount of processing involved cannot be trivially inferred. Furthermore, the applied vertex transform is to be accounted, as well topology information specific to the system being evaluated as post-transform caching can introduce consistent variations in the expected results.
Point in polygon test. In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x0,y0) lies inside a simple polygon given by a sequence of line segments. It is known as the Point in polygon test.

## In popular culture

• They Might Be Giants have a song entitled "Nonagon" on their children's album "Here Come the 123s." The song anthropomorphizes each of the regular polygons with three through eight sides (except the heptagon), placing them at a party hosted by the Nonagon. A video on the DVD featuring this song shows each of the polygons as equiangular shapes with simply-drawn human characteristics.[5]

## References

### Notes

1. ^ Polygon Area and Centroid
2. ^ A.M. Lopshits (1963). Computation of areas of oriented figures. D C Heath and Company: Boston, MA.
3. ^ Meditation VI by Descartes (English translation).
4. ^ Geometry Demystified: A Self-teaching Guide By Stan Gibilisco Published by McGraw-Hill Professional, 2003 ISBN 0071416501, 9780071416504
5. ^ http://tmbw.net/wiki/Lyrics:Nonagon

### Bibliography

• Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948).
• Cromwell, P.;Polyhedra, CUP hbk (1997), pbk. (1999).
• Grünbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and comput. geom: the Goodman-Pollack festschrift, ed. Aronov et al. Springer (2003) pp. 461–488. (pdf)

# Study guide

Up to date as of January 14, 2010

### From Wikiversity

 Run a search on Polygon at Wikipedia.
 Search Wikimedia Commons for images, sounds and other media related to: Polygon
 Search for Polygon on the following projects:
 Lost on Wikiversity? Please help by choosing project boxes to classify this resource by: subject educational level resource type
.A polygon is a plane figure that is bounded by a closed path or circuit.^ A polygon is a closed figure composed of three or more line segments that intersect at their endpoints.
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

.For a polygon with n sides, the sum of the measures of the polygon's angles is given by the equation ((n-2)180)/n.^ Question 254550 : how many sides does a polygon have if thhe sum of its interior angles is 540?
• Questions on Geometry: Polygons answered by real tutors! 12 January 2010 2:53 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.
• Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]

^ Question 252958 : the polygon is convex ,with sides a-g wat is the sum of its interior angle measures Answer by drk(669) ( Show Source ): You can put this solution on YOUR website!
• Questions on Geometry: Polygons answered by real tutors! 12 January 2010 2:53 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

# 1911 encyclopedia

Up to date as of January 14, 2010

### From LoveToKnow 1911

POLYGON (Gr. iroXis, many, and .ywvia, an angle), in geometry, a figure enclosed by any number of lines - the sides - which intersect in pairs at the corners or vertices.^ Angle: A figure formed by the intersection of two sides.
• Polygon - examples, body, used, form, parts, Terminology used in describing polygons, Examples of polygons 12 January 2010 2:53 UTC www.scienceclarified.com [Source type: Reference]

^ The vertices of a polygon are the points where its sides intersect.
• Figures and polygons 12 January 2010 2:53 UTC www.mathleague.com [Source type: Reference]

^ The number of sides The interior angles (the angles inside).

.If the sides are coplanar, the polygon is said to be "plane"; if not, then it is a "skew" or "gauche" polygon.^ As jkd said, one and two sided polygons exist in some non Euclidean geometries.
• Can you name the polygons shown below? - sporcle 12 January 2010 2:53 UTC www.sporcle.com [Source type: General]

^ A polygon is said to be regular if and only if the following conditions are satisfied: (1) All the sides are of equal length.
• Regular Polygon 11 September 2009 9:28 UTC www.welltall.com [Source type: Academic]

^ If the sides are coplanar, the polygon is said to be "plane"; if not, then it is a "skew" or "gauche" polygon.

.If the figure lies entirely to one side of each of the bounding lines the figure is "convex"; << " if not it is re-entrant or concave.^ Figure 5 Figure 5 shows the case of a polygon in which one of its sides lies entirely on the threshold.
• Determining Whether A Point Is Inside A Complex Polygon 12 January 2010 2:53 UTC alienryderflex.com [Source type: FILTERED WITH BAYES]

^ A figure that is not convex is called a concave figure.
• Figures and polygons 12 January 2010 2:53 UTC www.mathleague.com [Source type: Reference]
• Polygons 12 January 2010 2:53 UTC www.gltech.org [Source type: Original source]

^ Note that on one side of the line, and on the other.

.A regular polygon has all its sides and angles equal, i.e. it is equilateral and equiangular; if the sides and angles be not equal the polygon is "irregular ."^ Polygons can have three personality characteristics: equilateral, equiangular, and regular.
• Sizing Up the Area of a Polygon - For Dummies 12 January 2010 2:53 UTC www.dummies.com [Source type: General]

^ Regular polygons are equiangular as well as equilateral.
• Polygon Properties - Tables and Formulas 12 January 2010 2:53 UTC www.ricksmath.com [Source type: Reference]

^ All the interior angles are equal.
• Regular Polygon 11 September 2009 9:28 UTC www.welltall.com [Source type: Academic]

.Of polygons inscriptible in a circle an equilateral figure is necessarily equiangular, but the converse is only true when the number of sides is odd.^ Regular polygons are equiangular as well as equilateral.
• Polygon Properties - Tables and Formulas 12 January 2010 2:53 UTC www.ricksmath.com [Source type: Reference]

^ A regular polygon is both equilateral and equiangular.
• Each Interior Angle 12 January 2010 2:53 UTC www.regentsprep.org [Source type: Original source]

^ Multiply that number by the number of sides of the polygon.
• POLYGONS MADE TO ORDER 12 January 2010 2:53 UTC www.iit.edu [Source type: Reference]

.The term regular polygon is usually restricted to "convex" polygons; a special class of polygons (regular in the wider sense) has been named "star polygons" on account of their resemblance to star-rays; these are, however, concave.^ A concave polygon is a polygon that is not convex.

^ We examine these concepts for regular polygons.
• Areas and Perimeters of Regular Polygons 12 January 2010 2:53 UTC www.algebralab.org [Source type: Academic]

^ Regular polygons are always convex.
• Polygon definition - Math Open Reference 12 January 2010 2:53 UTC www.mathopenref.com [Source type: FILTERED WITH BAYES]

.Polygons, especially of the "regular" and "star" types, were extensively studied by the Greek geometers.^ Polygons, especially of the "regular" and "star" types, were extensively studied by the Greek geometers.

^ The existence of many types of regular polygons is in contrast with the case for regular polyhedra, of which only five distinct types in Euclidean geometry .
• PlanetMath: regular polygon 12 January 2010 2:53 UTC planetmath.org [Source type: FILTERED WITH BAYES]

^ The term regular polygon is usually restricted to "convex" polygons; a special class of polygons (regular in the wider sense) has been named " star polygons" on account of their resemblance to star-rays; these are, however, concave.

.There are two important corollaries to prop.^ There are two important corollaries to prop.

.32, book i., of Euclid's Elements relating to polygons.^ Euclid's Elements relating to polygons.

^ The systematic discussion of regular polygons with respect to the inscribed and circumscribed circles is given in the fourth book of the Elements.

^ The methods of Euclid permit the construction of the following series of inscribed polygons: from the square, the 8-side or octagon, 16-, 32-..

.Having proved that the sum of the angles of a triangle is a straight angle, i.e. two right angles, it is readily seen that the sum of the internal angles of a polygon (necessarily convex) of n sides is n -2 straight angles (2n-4 right angles), for the polygon can be divided into n2 triangles by lines joining one vertex to the other vertices.^ Triangles are three-sided polygons.
• Activity 4 12 January 2010 2:53 UTC homepage.mac.com [Source type: FILTERED WITH BAYES]

^ A diagonal of a polygon is a line segment that connects two vertices of the polygon that are not next to each other.
• Activity 4 12 January 2010 2:53 UTC homepage.mac.com [Source type: FILTERED WITH BAYES]

^ The perimeter of a polygon is the sum of the lengths of its sides.
• PlanetMath: polygon 11 September 2009 9:28 UTC planetmath.org [Source type: FILTERED WITH BAYES]

.The second corollary is that the sum of the supplements of the internal angles, measured in the same direction, is 4 right angles, and is thus independent of the number of sides.^ An important property of polygons is that all polygons with the same number of sides have the same sum of interior angle s.
• Totally Tessellated: Polygons and Angles 12 January 2010 2:53 UTC library.thinkquest.org [Source type: FILTERED WITH BAYES]

^ Vary the number of sides and determine how the sum of the angles changes.
• Polygon Angle Sum - Activity A Gizmo | ExploreLearning 12 January 2010 2:53 UTC www.explorelearning.com [Source type: General]

^ Regular polygons are polygons that have all sides the same length and all angles the same measure.
• Activity 4 12 January 2010 2:53 UTC homepage.mac.com [Source type: FILTERED WITH BAYES]

.The systematic discussion of regular polygons with respect to the inscribed and circumscribed circles is given in the fourth book of the Elements. (We may note that the construction of an equilateral triangle and square appear in the first book.^ Regular: A polygon that is both equilateral and equiangular.
• Polygon - examples, body, used, form, parts, Terminology used in describing polygons, Examples of polygons 12 January 2010 2:53 UTC www.scienceclarified.com [Source type: Reference]

^ Stages in the construction of a regular polygon.

^ Regular polygons are equiangular as well as equilateral.
• Polygon Properties - Tables and Formulas 12 January 2010 2:53 UTC www.ricksmath.com [Source type: Reference]

) .The triangle is discussed in props.^ The triangle is discussed in props.

2-6; the square in props. 6-9; the pentagon (5-side) in props. .Io-14; the hexagon (6-side) in prop.^ Io-14; the hexagon (6-side) in prop.

15; and the quindecagon in prop. .16. The triangle and square call for no special mention here, other than that any triangle can be inscribed or circumscribed to a circle.^ Construct a square inscribed in a circle.
• POLYGONS MADE TO ORDER 12 January 2010 2:53 UTC www.iit.edu [Source type: Reference]

^ Below is a square with its inscribed circle drawn in green and its circumscribed circle drawn in cyan.
• PlanetMath: regular polygon 12 January 2010 2:53 UTC planetmath.org [Source type: FILTERED WITH BAYES]

^ The triangle and square call for no special mention here, other than that any triangle can be inscribed or circumscribed to a circle.

.The pentagon is of more interest.^ The pentagon is of more interest.

.Euclid bases his construction upon the fact that the isosceles triangle formed by joining the extremities of one side of a regular pentagon to the opposite vertex has each angle at the base double the angle at the vertex.^ Vertex: A point where any two of the sides of a polygon meet to form an angle.
• Polygon - examples, body, used, form, parts, Terminology used in describing polygons, Examples of polygons 12 January 2010 2:53 UTC www.scienceclarified.com [Source type: Reference]

^ The regular pentagon has five lines of symmetry: one between each vertex and its opposite side.
• Totally Tessellated: Regular Polygons 11 September 2009 9:28 UTC library.thinkquest.org [Source type: Original source]

^ Join the extremities of one side to the opposite vertex, and consider the triangle so formed.

.He constructs this triangle in prop.^ He constructs this triangle in prop.

^ The inscription of a pentagon in a circle is effected by inscribing an isosceles triangle similar to that constructed in prop.

to, by dividing a line in medial section, .i.e. the square of one part equal to the product of the other part and the whole line (a construction given in book ii.^ Reviews of books and other products .
• Polygon (math) - PHP Classes 12 January 2010 2:53 UTC www.phpclasses.org [Source type: Reference]

^ The symbol , which includes the arrow heads at both ends indicates the whole line where , which does not have the arrow heads, indicates a line segment, which is finite in length (only the part of the line from A to B).
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ Some polygons, especially convex ones, are monotone with respect to several lines, while other polygons, like the one you see above, is monotone with respect to the vertical only.
• Polygon Partitioning: Monotone Triangulation 12 January 2010 2:53 UTC www.me.cmu.edu [Source type: FILTERED WITH BAYES]

.II), and then showing that the greater segment is the base of the required triangle, the remaining sides being each equal to the whole line.^ A triangle with three equal sides is an equilateral triangle .
• Definitions and Polygons 12 January 2010 2:53 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

^ II), and then showing that the greater segment is the base of the required triangle, the remaining sides being each equal to the whole line.

^ An isosceles triangle is a triangle that has two equal sides: .
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

.The inscription of a pentagon in a circle is effected by inscribing an isosceles triangle similar to that constructed in prop.^ He constructs this triangle in prop.

^ The inscription of a pentagon in a circle is effected by inscribing an isosceles triangle similar to that constructed in prop.

^ Construct a square inscribed in a circle.
• POLYGONS MADE TO ORDER 12 January 2010 2:53 UTC www.iit.edu [Source type: Reference]

to, bisecting the angles at the base and producing the bisectors to meet the circle. .Euclid then proves that these intersections and the three vertices of the triangle are the vertices of the required pentagon.^ Euclid then proves that these intersections and the three vertices of the triangle are the vertices of the required pentagon.

^ A pentagon has 5 sides, and can be made from three triangles , so you know what ...
• Interior Angles of Polygons 11 September 2009 9:28 UTC www.mathsisfun.com [Source type: Original source]

^ However for a pentagon close to a triangle, with two vertices close to one vertex on the bottom and two close to the other vertex on the bottom, the area of the midpoint figure is as close to 3/4 as we please (see demo 3 below).
• Midpoint Polygons | Convex Pentagons 12 January 2010 2:53 UTC techhouse.brown.edu [Source type: FILTERED WITH BAYES]

.The circumscription of a pentagon is effected by constructing an inscribed pentagon, and drawing tangents to the circle at the vertices.^ The inscription of a pentagon in a circle is effected by inscribing an isosceles triangle similar to that constructed in prop.

^ Construct a square inscribed in a circle.
• POLYGONS MADE TO ORDER 12 January 2010 2:53 UTC www.iit.edu [Source type: Reference]

^ The circumscription of a pentagon is effected by constructing an inscribed pentagon, and drawing tangents to the circle at the vertices.

.This supplies a general method for circumscribing a polygon if the inscribed be given, and conversely.^ This supplies a general method for circumscribing a polygon if the inscribed be given, and conversely.

^ Generating random polygons with given vertices.
• Generating random simple polygons 12 January 2010 2:53 UTC compgeom.cs.uiuc.edu [Source type: FILTERED WITH BAYES]

^ Mensuration .-In the regular polygons the fact that they can be inscribed and circumscribed to a circle affords convenient expressions for their area, &c.

.In book xiii., prop.^ In book xiii., prop.

.to, an alternative method for inscribing a pentagon is indicated, for it is there shown that the sum of the squares of the sides of a square and hexagon inscribed in the same circle equals the square of the side of the pentagon.^ The side of a hexagon inscribed in a circle obviously equals the radius of the circle.

^ A regular hexagon has got 6 sides of equal length .

^ Everyday Pentagons Hexagon A hexagon has six sides.
• Polygons powerpoint 12 January 2010 2:53 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

.It may be incidentally noticed that Euclid's construction of the isosceles triangle which has its basal angles double the vertical angle solves the problem of quinquesecting a right angle; moreover, the base of the triangle is the side of the regular decagon inscribed in a circle having the vertex as centre and the sides of the triangle as radius.^ The side of a hexagon inscribed in a circle obviously equals the radius of the circle.

^ A triangle having an obtuse angle.
• Figures and polygons 12 January 2010 2:53 UTC www.mathleague.com [Source type: Reference]

^ To construct a regular polygon inscribed in a circle by using isosceles triangles with vertex angles at the center of the circle and legs as radii.
• POLYGONS MADE TO ORDER 12 January 2010 2:53 UTC www.iit.edu [Source type: Reference]

.The inscription of a hexagon in a circle (prop.^ The inscription of a hexagon in a circle (prop.

^ The inscription of a pentagon in a circle is effected by inscribing an isosceles triangle similar to that constructed in prop.

.15) reminds one of the Pythagorean result that six equilateral triangles placed about a common vertex form a plane; hence the bases form a regular hexagon.^ The equilateral triangle is one of the three regular polygons that tile a plane.
• Geometry in Art & Architecture Unit 5 12 January 2010 2:53 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

^ There are three regular tessellations of the plane: by triangles, by squares, by hexagons.
• Tessellations by Polygons - EscherMath 12 January 2010 2:53 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ Pythagorean result that six equilateral triangles placed about a common vertex form a plane; hence the bases form a regular hexagon.

.The side of a hexagon inscribed in a circle obviously equals the radius of the circle.^ The side of a hexagon inscribed in a circle obviously equals the radius of the circle.

^ A regular hexagon has got 6 sides of equal length .

^ Regular polygons can be inscribed by a circle such that the circle is tangent to the sides at the centers, and circumscribed by a circle such that the sides form chords of the circle.
• Classifying Polygons - Watch video (Geometry) 12 January 2010 2:53 UTC www.winpossible.com [Source type: General]

.The inscription of the quindecagon in a circle is made to depend upon the fact that the difference of the arcs of a circle intercepted by covertical sides of a regular pentagon and equilateral triangle is 3 = of the whole circumference, and hence the bisection of this intercepted arc (by book iii., 30) gives the side of the quindecagon.^ The inscription of the quindecagon in a circle is made to depend upon the fact that the difference of the arcs of a circle intercepted by covertical sides of a regular pentagon and equilateral triangle is 3 = of the whole circumference, and hence the bisection of this intercepted arc (by book iii., 30) gives the side of the quindecagon.

^ Inscribe an equilateral triangle in the circle, the three corners all on the circles circumference.
• Puzzle - Nested Polygons 12 January 2010 2:53 UTC olimu.com [Source type: FILTERED WITH BAYES]

^ Depending on the angles and the sides we can sort the triangles into different types.
• Squares, Rectangles, Parallelograms and Other Polygons - EscherMath 12 January 2010 2:53 UTC euler.slu.edu [Source type: Original source]

.The methods of Euclid permit the construction of the following series of inscribed polygons: from the square, the 8-side or octagon, 16-, 32-..^ Yes, a square is a 4-sided regular polygon.
• 2D geometric objects: Polygons, circles. 12 January 2010 2:53 UTC faculty.matcmadison.edu [Source type: FILTERED WITH BAYES]

^ A eight-sided polygon is called a octagon .
• Definitions and Polygons 12 January 2010 2:53 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

^ Construct a regular octagon An octagon is an eight-sided polygon.

., or generally .4.2 n -side; from the hexagon, the 12-side or dodecagon, 24-, 48- ., or generally the 6.2 n -side; from the pentagon, the ro-side or decagon, 20-, 40-..^ For larger numbers of sides, one uses a Greek number prefix, as in regular pentagon'' and regular hexagon''.
• PlanetMath: regular polygon 12 January 2010 2:53 UTC planetmath.org [Source type: FILTERED WITH BAYES]

^ The words for polygons with sides (e.g., pentagon , hexagon , heptagon , etc.
• Regular Polygon -- from Wolfram MathWorld 12 January 2010 2:53 UTC mathworld.wolfram.com [Source type: Academic]

^ DrDeth Nov 20 2008 at 12:40 PM .
• Farseer Physics Engine - Discussions - Polygon Operations - Feedback / Progress 12 January 2010 2:53 UTC www.codeplex.com [Source type: General]

., or generally 5.2 n side; from the quindecagon, the 30-, 60- ., or generally 15.2 n - side. It was long supposed that no other inscribed polygons were possible of construction by elementary methods (i.e. by the ruler and compasses); Gauss disproved this by forming the 17-side, and he subsequently generalized his method for the (2 n +1)-side, when this number is prime.
.The problem of the construction of an inscribed heptagon, nonagon, or generally of any polygon having an odd number of sides, is readily reduced to the construction of a certain isosceles triangle.^ A general polygon with n sides can be cut into n − 2 triangles and so we have: .
• Tessellations by Polygons - EscherMath 12 January 2010 2:53 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ The problem of the construction of an inscribed heptagon, nonagon, or generally of any polygon having an odd number of sides, is readily reduced to the construction of a certain isosceles triangle.

^ A polygon’s name reflects the number of sides it has.
• Polygon - MSN Encarta 11 September 2009 9:28 UTC encarta.msn.com [Source type: General]

.Suppose the polygon to have (2n+1) sides.^ Suppose the polygon to have ( 2n+1 ) sides.

^ And there are 2 such triangles per side, or 2n for the whole polygon : .
• Regular Polygons - Properties 11 September 2009 9:28 UTC www.mathsisfun.com [Source type: FILTERED WITH BAYES]

.Join the extremities of one side to the opposite vertex, and consider the triangle so formed.^ Join the extremities of one side to the opposite vertex, and consider the triangle so formed.

^ A segment is a median of a triangle if and only if it connects one vertex to the midpoint of the opposite side.
• Symmetry and Polygons 12 January 2010 2:53 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

^ Edge One side of a polygon or triangle.
• Polygon Modeling - Wikibooks, collection of open-content textbooks 12 January 2010 2:53 UTC en.wikibooks.org [Source type: FILTERED WITH BAYES]

.It is readily seen that the angle at the base is n times the angle at the vertex.^ It is readily seen that the angle at the base is n times the angle at the vertex.

^ Euclid bases his construction upon the fact that the isosceles triangle formed by joining the extremities of one side of a regular pentagon to the opposite vertex has each angle at the base double the angle at the vertex.

^ The side opposite the vertex angle is called the base .
• Symmetry and Polygons 12 January 2010 2:53 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

.In the heptagon the ratio is 3, in the nonagon 4, and so on.^ In the heptagon the ratio is 3, in the nonagon 4, and so on.

The Arabian geometers of the 9th century showed that the heptagon required the solution of a cubic equation, thus resembling the Pythagorean problems of "duplicating the cube" and "trisecting an angle." Edmund Halley gave solutions for the heptagon and nonagon by means of the parabola and circle, and by a parabola and hyperbola respectively.
.Although rigorous methods for inscribing the general polygons in a circle are wanting, many approximate ones have been devised.^ A polygon is inscribed in a circle if each vertex of the polygon is a point on the circle.
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ Although rigorous methods for inscribing the general polygons in a circle are wanting, many approximate ones have been devised.

^ Circle inscribingCircle () Return the inscribing circle of this polygon.

.Two such methods are here given: (I) Divide the diameter of the circle into as many parts as the polygon has sides.^ What could be inside a two-sided polygon?

^ How many sides does the polygon have ?
• Questions on Geometry: Polygons answered by real tutors! 12 January 2010 2:53 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ How many sides does the polygon have?
• Questions on Geometry: Polygons answered by real tutors! 12 January 2010 2:53 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

.On the diameter construct an equilateral triangle; and from its vertex draw a line through the second division along the diameter, measured from an extremity, and produce this line to intercept the circle.^ The measure of the central angles of an equilateral triangle: .
• Cool math .com - Polygons - properties, interior angles, triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons, decagons, 11-gons, dodecagons 12 January 2010 2:53 UTC www.coolmath.com [Source type: Original source]

^ These return vertices are notches for which both incident edges are on the same side of the vertical line that passes through their common vertex.

^ Using a protractor and a ruler, draw a line, at an angle a to the first line, from O to the circumference of the circle.

.Then the chord joining this point to the extremity of the diameter is the side of the required polygon.^ The vertices of a polygon are the points where its sides intersect.
• Figures and polygons 12 January 2010 2:53 UTC www.mathleague.com [Source type: Reference]

^ It is important to know that each point of a polygon at which two sides cross each other is called a vertex.
• Interior Angles of a Polygon - Free Math Help 12 January 2010 2:53 UTC www.freemathhelp.com [Source type: FILTERED WITH BAYES]

^ Testing the Algorithm The test program (Listing 2) draws a random 40-sided polygon and then picks random points to throw at the inpoly() routine.

.(2) Divide the diameter as before, and draw also the perpendicular diameter.^ Divide the diameter as before, and draw also the perpendicular diameter.

^ Join the points so obtained; and draw a line from the point nearest the divided diameter where this line intercepts the circle to the third division from the produced extremity; this line is the required length.

.Take points on these diameters beyond the circle and at a distance from the circle equal to one division of the diameter.^ Take points on these diameters beyond the circle and at a distance from the circle equal to one division of the diameter.

^ Considering the case of points it is obvious that we can join a chosen point with any one of the remaining (n I) points; any one of these (n - I) points can be joined to any one of the remaining (n-2), and by proceeding similarly it is seen that we can pass through the n points in (n - I) (n -2).

^ The circles with ones in them represent points counted once if the snail crosses them.

.Join the points so obtained; and draw a line from the point nearest the divided diameter where this line intercepts the circle to the third division from the produced extremity; this line is the required length.^ On the diameter construct an equilateral triangle; and from its vertex draw a line through the second division along the diameter, measured from an extremity, and produce this line to intercept the circle.

^ Divide the diameter as before, and draw also the perpendicular diameter.

^ Then the chord joining this point to the extremity of the diameter is the side of the required polygon.

.The construction of any regular polygon on a given side may be readily performed with a protractor or scale of chords, for it is only necessary to lay off from the extremities of the given side lines equal in length to the given base, at angles equal to the interior angle of the polygon, and repeating the process at each extremity so obtained, the angle being always taken on the same side; or lines may be laid off at one half of the interior angles, describing a circle having the meet of these lines as centre and their length as radius, and then measuring the given base around the circumference.^ A polygon may be regarded as determined by the joins of points or the meets of lines.

^ Because the polygon is regular, the lengths are the same for each side.
• Sizing Up the Area of a Polygon - For Dummies 12 January 2010 2:53 UTC www.dummies.com [Source type: General]

^ The side of a hexagon inscribed in a circle obviously equals the radius of the circle.

.Star Polygons.-These figures were studied by the Pythagoreans, and subsequently engaged the attention of many geometersBoethius, Athelard of Bath, Thomas Bradwardine, archbishop of Canterbury, Johannes Kepler and others.^ Star Polygons.-These figures were studied by the Pythagoreans, and subsequently engaged the attention of many geometersBoethius, Athelard of Bath , Thomas Bradwardine , archbishop of Canterbury , Johannes Kepler and others.

^ The term regular polygon is usually restricted to "convex" polygons; a special class of polygons (regular in the wider sense) has been named " star polygons" on account of their resemblance to star-rays; these are, however, concave.

^ Mystical and magical properties were assigned to them at an early date; the Pythagoreans regarded the pentagram, the star polygon derived from the pentagon, as the symbol of health, the Platonists of well-being, while others used it to symbolize happiness.

.Mystical and magical properties were assigned to them at an early date; the Pythagoreans regarded the pentagram, the star polygon derived from the pentagon, as the symbol of health, the Platonists of well-being, while others used it to symbolize happiness.^ A view in another direction may well show the 5-metre polygons that Mellon and other scientists were expecting.
• Mars scientists ponder polygon mystery - space - 27 May 2008 - New Scientist 12 January 2010 2:53 UTC www.newscientist.com [Source type: General]

^ The general formula for the area of the midpoint polygon is (1/2)(Area of the Original Polygon) + (1/4)(Area of the Star Polygon), where the star polygon is constructed from the original by taking every other vertex in order.
• Midpoint Polygons | Convex Pentagons 12 January 2010 2:53 UTC techhouse.brown.edu [Source type: FILTERED WITH BAYES]

^ The polygon reducer, as well as many other 3D mesh processing tools, can also be enabled during batch conversion of 3D files.
• Okino's Polygon Reduction System 12 January 2010 2:53 UTC www.okino.com [Source type: Reference]

.Engraven on metal, &c., it is worn in almost every country as a charm or amulet.^ Engraven on metal , &c., it is worn in almost every country as a charm or amulet .

.The pentagon gives rise to one star polygon, the hexagon gives none, the heptagon two, the octagon one, and the nonagon two.^ Pentagon Hexagon Heptagon Octagon That does not seem too many does it?

^ The pentagon gives rise to one star polygon, the hexagon gives none, the heptagon two, the octagon one, and the nonagon two.

^ Tetragon, 4 sides Pentagon, 5 sides Hexagon, 6 sides Heptagon, 7 sides Octagon, 8 sides Nonagon Enneagon, 9 sides Decagon, 10 sides Undecagon, 11 sides Dodecagon, 12 sides (C) 2009 Copyright John Page .
• Polygon definition - Math Open Reference 12 January 2010 2:53 UTC www.mathopenref.com [Source type: FILTERED WITH BAYES]

.In general, the number of star polygons which can be drawn with the vertices of an n-point regular polygon is the number of numbers which are not factors of n and are less than In.^ In any regular polygon, a point termed the center is equidistant from all vertices.
• Symmetry and Polygons 12 January 2010 2:53 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

^ If it's a star or a polygon, it's a point on the radius and the centre.
• Scratch | Projects tagged with 'polygon' 12 January 2010 2:53 UTC scratch.mit.edu [Source type: FILTERED WITH BAYES]

^ The center of a regular polygon is the point that is equidistant from each of its vertices .
• PlanetMath: regular polygon 12 January 2010 2:53 UTC planetmath.org [Source type: FILTERED WITH BAYES]

Pentagrams. Heptagrams. Nonograms.
.Number of n-point and n-side Polygons. A polygon may be regarded as determined by the joins of points or the meets of lines.^ The vertices of a polygon are the points where its sides intersect.
• Figures and polygons 12 January 2010 2:53 UTC www.mathleague.com [Source type: Reference]

^ Polygons are named by the number of sides they have.
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners.
• RF Cafe - Geometry Polygons 12 January 2010 2:53 UTC www.rfcafe.com [Source type: Academic]

The termination -gram is often applied to the figures determined by lines, e.g. pentagram, hexagram. .It is of interest to know how many polygons can be formed with n given points as vertices (no three of which are collinear), or with n given lines as sides (no two of which are parallel).^ How many diagonals does an n -sided polygon have?

^ The points and the segments are known as vertices and sides of the polygon.
• Regular Polyhedra from Interactive Mathematics Miscellany and Puzzles 12 January 2010 2:53 UTC www.cut-the-knot.org [Source type: FILTERED WITH BAYES]

^ Quadrilaterals can be classify by the lengths of their sides and how many pairs of sides are parallel.
• Symmetry and Polygons 12 January 2010 2:53 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

.Considering the case of points it is obvious that we can join a chosen point with any one of the remaining (n I) points; any one of these (n - I) points can be joined to any one of the remaining (n-2), and by proceeding similarly it is seen that we can pass through the n points in (n - I) (n -2).^ These return vertices are notches for which both incident edges are on the same side of the vertical line that passes through their common vertex.

^ This means that these methods consider unclosed shapes to be implicitly closed for the purpose of determining if a shape contains or intersects a rectangle or if a shape contains a point.
• Shape (Java 2 Platform SE v1.4.2) 11 September 2009 9:28 UTC java.sun.com [Source type: Reference]

^ A polygonal chain C , is considered to be monotonic with respect to a line L , if every line, L` , which is orthogonal to L , meets C in at most one point.
• Polygon Partitioning: Monotone Triangulation 12 January 2010 2:53 UTC www.me.cmu.edu [Source type: FILTERED WITH BAYES]

.. 2.1 or (n-I)! ways. It is obvious that the direction in which we pass is immaterial; hence we must divide this number by 2, thus obtaining (n-I)!/2 as the required number. .In a similar manner it may be shown that the number of polygons determined by n lines is (n - I) !/2. Thus five points or lines determine 12 pentagons, 6 points or lines 60 hexagons, and so on.^ A point on the polygon line is considered in.
• Class: Polygon 12 January 2010 2:53 UTC resources.esri.com [Source type: FILTERED WITH BAYES]

^ The polygon has even-odd winding, meaning that a point is inside the shape if it crosses the boundary an odd number of times on the way to infinity.
• Polygon (GNU Classpath 0.95 Documentation) 12 January 2010 2:53 UTC developer.classpath.org [Source type: FILTERED WITH BAYES]

^ In a valid polygon, holes may touch the shell or other holes at a single point.

 Number of sides. 3 Triangle. 4 Square. 5 Pentagon. 6 Hexagon. 7 Heptagon. 8 Octagon. 9 Nonagon. 10 Decagon. 11 Undecagon. 12 Dodecagon. a 60° 90° 108° 1200 1284° 135° 140° 144° 1471i 150° 0 120° 90° 72° 60° 51'° 45° 40° 36° 321i° 30° A 0.43301 I 1.72048 2.59808 3.63391 4.82843 6.18182 7.69421 9.36564 11.19615 R 0.57735 0.70710 0.85065 I 1.1523 1.3065 1.4619 I.6180 1.7747 1.9318 r 0.28867 0.5 0.688,9 0.86602 1.0383 1.2071 1.3737 1.5388 1.7028 1.8660
.Mensuration.-In the regular polygons the fact that they can be inscribed and circumscribed to a circle affords convenient expressions for their area, &c.^ If we circumscribe the polygon in a circle, what is the radius of that circle?

^ A polygon is inscribed in a circle if each vertex of the polygon is a point on the circle.
• Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ If and are the perimeters of the regular polygons inscribed in and circumscribed around a given circle and and their areas, then .
• Regular Polygon -- from Wolfram MathWorld 12 January 2010 2:53 UTC mathworld.wolfram.com [Source type: Academic]

.In a n-gon, i.e. a polygon with n-sides, each side subtends at the centre the angle 27r/n, i.e. 360°/n, and each internal angle is (n-2)mr/n or (n-2) 180°/n.^ Question 254550 : how many sides does a polygon have if thhe sum of its interior angles is 540?
• Questions on Geometry: Polygons answered by real tutors! 12 January 2010 2:53 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

^ Summary: In Section I, students derive formulas for the angles, perimeter, radius, apothem, area, median length, and diagonal lengths of a regular polygon with n sides.

^ Corners of the tiles need to fit together around a point, which means the corner angle of the regular polygon must evenly divide 360°.
• Tessellations by Polygons - EscherMath 12 January 2010 2:53 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

.Calling the length of side a we may derive the following relations: Area (A) = 4 a 2 n cot (lr/n); radius of circum-circle (R) = 2 a cosec (lr/n) radius of in-circle (r) = 2a cot (lr/n).^ Summary: In Section I, students derive formulas for the angles, perimeter, radius, apothem, area, median length, and diagonal lengths of a regular polygon with n sides.

^ The actual apothem is 1/2 sqr(25 + 10*(sqr(5)), or about 3.44 The actual area for a regular pentagon of side length 5 is just over 43.
• Sizing Up the Area of a Polygon - For Dummies 12 January 2010 2:53 UTC www.dummies.com [Source type: General]

^ Not only the vertices may overlap, but the sides should be permitted to have zero length.
• Polygons: formality and intuition. Polygonal metamorphosis. from Interactive Mathematics Miscellany and Puzzles 12 January 2010 2:53 UTC www.cut-the-knot.org [Source type: FILTERED WITH BAYES]

The table at foot of p. .1592 gives the value of the internal angle (a), the angle # subtended at the centre by a side, area (A), radius of the circum-circle (R), radius of the inscribed circle (r) for the simpler polygons, the length of the side being taken as unity.^ A regular polygon is a polygon in which every side has the same length and every angle is the same.

^ If we circumscribe the polygon in a circle, what is the radius of that circle?

^ If a regular polygon has n sides, and each of the sides has length s , what is its perimeter p ?

# Wiktionary

Up to date as of January 15, 2010

Polygon n.

# Simple English

A polygon is a closed two-dimensional shape. It usually has three sides/corners or more. It could also be referred to as 'A closed plane figure bound by three or more line segments'. It has a number of edges. These edges are connected by lines. A square is a polygon because it has four sides. The smallest possible polygon in a Euclidean geometry or "flat geometry" is the triangle, but on a sphere, there can be a digon. The monogon is a theoretical figure that cannot exist - it has only one side and one edge.

If the edges (lines of the polygon) do not intersect (cross each other) , the polygon is called simple, otherwise it is complex.

In computer graphics, polygons (especially triangles) are often used to make graphics.

# Citable sentences

Up to date as of December 21, 2010

Here are sentences from other pages on Polygon, which are similar to those in the above article.