# Area: Wikis

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# Encyclopedia

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of surfaces.[1]

## Units

Units for measuring area, with exact conversions, include:
• square metre (m2)
• are (a) = 100 square metres (m2)
• hectare (ha) = 100 ares = 10000 square metres
• square kilometre (km2) = 100 hectares = 10000 ares = 1000000 square metres
• square megametre (Mm2) = 1000000000000 square metres
• square foot = 144 square inches = 0.09290304 square metres
• square yard = 9 square feet = 0.83612736 square metres
• square perch = 30.25 square yards = 25.2928526 square metres
• acre = 10 square chains = one furlong by one chain = 160 square perches = 4840 square yards = 43560 square feet = 4046.8564224 square metres
• square mile = 640 acres = 2.589988110336 square kilometres

## Formulae

Common formulae for area:
Shape Formula Variables
Regular triangle (equilateral triangle) $frac14\sqrt{3}s^2\,\!$ s is the length of one side of the triangle.
Triangle $\sqrt{s(s-a)(s-b)(s-c)}\,\!$ s is half the perimeter, a, b and c are the length of each side.
Triangle $frac12 a b \sin(C)\,\!$ a and b are any two sides, and C is the angle between them.
Triangle $frac12bh \,\!$ b and h are the base and altitude (measured perpendicular to the base), respectively.
Square $s^2\,\!$ s is the length of one side of the square.
Rectangle $lw \,\!$ l and w are the lengths of the rectangle's sides (length and width).
Rhombus $frac12ab$ a and b are the lengths of the two diagonals of the rhombus.
Parallelogram $bh\,\!$ b is the length of the base and h is the perpendicular height.
Trapezoid $frac12(a+b)h \,\!$ a and b are the parallel sides and h the distance (height) between the parallels.
Regular hexagon $frac32\sqrt{3}s^2\,\!$ s is the length of one side of the hexagon.
Regular octagon $2\left(1+\sqrt{2}\right)s^2\,\!$ s is the length of one side of the octagon.
Regular polygon $\frac{ns^2} {4 \cdot an(\pi/n)}\,\!$ s is the sidelength and n is the number of sides.
$frac12a p \,\!$ a is the apothem, or the radius of an inscribed circle in the polygon, and p is the perimeter of the polygon.
Circle $\pi r^2\ ext{or}\ \frac{\pi d^2}{4} \,\!$ r is the radius and d the diameter.
Circular sector $frac12 r^2 heta \,\!$ r and θ are the radius and angle (in radians), respectively.
Ellipse $\pi ab \,\!$ a and b are the semi-major and semi-minor axes, respectively.
Total surface area of a Cylinder $2\pi r (r + h)\,\!$ r and h are the radius and height, respectively.
Lateral surface area of a cylinder $2 \pi r h \,\!$ r and h are the radius and height, respectively.
Total surface area of a Cone $\pi r (r + l) \,\!$ r and l are the radius and slant height, respectively.
Lateral surface area of a cone $\pi r l \,\!$ r and l are the radius and slant height, respectively.
Total surface area of a Sphere $4\pi r^2\ ext{or}\ \pi d^2\,\!$ r and d are the radius and diameter, respectively.
Total surface area of an ellipsoid   See the article.
Square to circular area conversion $\frac{4}{\pi} A\,\!$ A is the area of the square in square units.
Circular to square area conversion $\frac{1}{4} C\pi\,\!$ C is the area of the circle in circular units.
.The above calculations show how to find the area of many common shapes.^ If you need to find common values for many types then FindCommonAttributes is much more efficient.
• Camelot: Range Class Reference 18 September 2009 17:13 UTC downloads.xara.com [Source type: FILTERED WITH BAYES]

The area of irregular polygons can be calculated using the "Surveyor's formula".[2]

### Areas of 2-dimensional figures

• a triangle: $frac12Bh$ (where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known. If the lengths of the three sides are known then Heron's formula can be used: $\sqrt{s(s-a)(s-b)(s-c)}$(where a, b, c are the sides of the triangle, and $s = frac12(a + b + c)$ is half of its perimeter) If an angle and its two included sides are given, then area=absinC where C is the given angle and a and b are its included sides. If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of (x1y2+ x2y3+ x3y1 - x2y1- x3y2- x1y3) all divided by 2. This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points, (x1,y1) (x2,y2) (x3,y 3). The shoelace formula can also be used to find the areas of other polygons when their vertices are known. .Another approach for a coordinate triangle is to use Infinitesimal calculus to find the area.
• a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points: $i + \frac{b}{2} - 1$, where i is the number of grid points inside the polygon and b is the number of boundary points.^ The main reason given for using the Beltway is that it saves time getting from one point to another.
• Final Report Capital Beltway Update: Beltway User Focus Groups 15 September 2009 4:04 UTC www.nhtsa.dot.gov [Source type: FILTERED WITH BAYES]

This result is known as Pick's theorem.

### Area in calculus

The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions
• the area between the graphs of two functions is equal to the integral of one function, f(x), minus the integral of the other function, g(x).
• an area bounded by a function r = r(θ) expressed in polar coordinates is ${1 \over 2} \int_0^{2\pi} r^2 \, d heta$.
• the area enclosed by a parametric curve $\vec u(t) = (x(t), y(t))$ with endpoints $\vec u(t_0) = \vec u(t_1)$ is given by the line integrals
$\oint_{t_0}^{t_1} x \dot y \, dt = - \oint_{t_0}^{t_1} y \dot x \, dt = {1 \over 2} \oint_{t_0}^{t_1} (x \dot y - y \dot x) \, dt$
or the z-component of
${1 \over 2} \oint_{t_0}^{t_1} \vec u imes \dot{\vec u} \, dt.$

### Surface area of 3-dimensional figures

• cube: 6s2, where s is the length of the top side
• rectangular box: $2 (\ell w + \ell h + w h)$ the length divided by height
• cone: $\pi r\left(r + \sqrt{r^2 + h^2}\right)$, where r is the radius of the circular base, and h is the height. That can also be rewritten as πr2 + πrl where r is the radius and l is the slant height of the cone. πr2 is the base area while πrl is the lateral surface area of the cone.
• prism: 2 × Area of Base + Perimeter of Base × Height

#### General formula

The general formula for the surface area of the graph of a continuously differentiable function z = f(x,y), where $(x,y)\in D\subset\mathbb{R}^2$ and D is a region in the xy-plane with the smooth boundary:
$A=\iint_D\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}\,dx\,dy.$
Even more general formula for the area of the graph of a parametric surface in the vector form $\mathbf{r}=\mathbf{r}(u,v),$ where $\mathbf{r}$ is a continuously differentiable vector function of $(u,v)\in D\subset\mathbb{R}^2$:
$A=\iint_D \left|\frac{\partial\mathbf{r}}{\partial u} imes\frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv.$[1]

## Area minimisation

Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles.
The question of the filling area of the Riemannian circle remains open.

## References

### Notes

1. ^ a b do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. Page 98.
2. ^ http://www.maa.org/pubs/Calc_articles/ma063.pdf

# 1911 encyclopedia

Up to date as of January 14, 2010
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# Wiktionary

Up to date as of January 15, 2010
(Redirected to area article)

## English

• help, file

From Latin area.

### Noun

Wikipedia has an article on:
 Singular area Plural areas or areæ
area (plural areas or areæ)
1. (mathematics) A measure of the extent of a planar region, or of the surface of a solid; it is measured in square units
2. A particular geographic region.
3. Any particular extent of surface.
The photo is a little dark in that area.
4. Figuratively, any extent, scope or range of an object or concept.
The plans are a bit vague in that area.
My guts are a bit sore in that area.
5. (British) An open space, below ground level, between the front of a house and the pavement

#### Translations

The translations below need to be checked and inserted above into the appropriate translation tables, removing any numbers. Numbers do not necessarily match those in definitions. See instructions at Help:How to check translations.

### Anagrams

• Anagrams of aaer
• æra

## Galician

### Noun

area f. (plural areas)
1. sand (grain)
2. sand (collectively)

#### Synonyms

• (sand collectively): xabre

## Italian

### Noun

area f. (plural aree)

### Anagrams

• Anagrams of aaer
• aera

## Latin

### Noun

ārea (genitive āreae); f, first declension
1. open space
2. a threshing floor
3. vocative singular of ārea
āreā f.
1. ablative singular of ārea

#### Inflection

Number Singular Plural
nominative ārea āreae
genitive āreae āreārum
dative āreae āreīs
accusative āream āreās
ablative āreā āreīs
vocative ārea āreae

area

## Swedish

### Noun

Inflection for area Singular Plural
common Indefinite Definite Indefinite Definite
Base form area arean areor areorna
Possessive form areas areans areors areornas
area c.
1. (geometry) area; a measure of squared distance.

# Genealogy

Up to date as of February 01, 2010

### From Familypedia

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of surfaces.

## Units

Units for measuring area include:
are (a) = 100 square meters (m²)
hectare (ha) = 100 ares (a) = 10000 square meters (m²)
square kilometre (km²) = 100 hectars (ha) = 10000 ares (a) = 1000000 square metres (m²)
square megametre (Mm²) = 1012 square metres
square foot = 144 square inches = 0.09290304 square metres (m²)
square yard = Template:Convert/sqft = 0.83612736 square metres (m²)
square perch = 30.25 square yards = 25.2928526 square metres (m²)
acre = 10 square chains or 160 square perches or 4840 square yards or Template:Convert/sqft = 4046.8564224 square metres (m²)
square mile = 640 acres (2.6 km2) = 2.5899881103 squarea kilometers (km²)
 This page uses content from the English language Wikipedia. The original content was at Area. The list of authors can be seen in the page history. As with this Familypedia wiki, the content of Wikipedia is available under the Creative Commons License.
 Wikipedia Area  +

# Simple English

Area is the amount of space a two dimensional (flat) surface takes up. It is useful because it is how much of a material is needed to make a hollow container; for example, how much wood is needed to make a wardrobe.

You can use different formulas to find the area of different shapes.

• Area of a rectangle is the length of any two touching sides multiplied together. In other words, length times width.
• Area of a triangle is half of the base multiplied by the perpendicular height. This can be found using the trigonometry formula, $A = 1/2 ab \sin c.$.
• Area of a circle: $A = \pi r^2$.

The area of a flat object is related to the surface area and volume of a three-dimensional object.

The area under a curve can be found using integration, from calculus.

Some units used to measure area are square mile and square kilometre.

# Citable sentences

Up to date as of December 08, 2010

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