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Encyclopedia

Updated live from Wikipedia, last check: February 04, 2012 03:35 UTC (36 seconds ago)

From Wikipedia, the free encyclopedia

This article is about the geometric quantity. For other uses, see Area (disambiguation).

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]

1 Units
2 Formulae
3 Additional formulae
4 Area minimisation
5 See also
6 References
- 6.1 Notes
7 External links

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.
Regular polygon	$frac12a p \,\!$	$a$ 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]

Additional formulae

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: , 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

$\oint_{t_0}^{t_1} x \dot y \, dt = - \oint_{t_0}^{t_1} y \dot x \, dt = {1 \over 2} \oint_{t_0}^{t_1} (x \dot y - y \dot x) \, dt$

(see Green's theorem)

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: $6 s 2$ , 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 $π r 2 + π r l$ where r is the radius and l is the slant height of the cone. $π r 2$ is the base area while $π r l$ 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

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

External links

Categories: Area

1911 encyclopedia

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Wiktionary

Up to date as of January 14, 2010

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Definition from Wiktionary, a free dictionary

See also área

English

Pronunciation

Audio (US)^{help, file}

Rhymes: -ɛəriə

Etymology

From Latin area.

Noun

Wikipedia has an article on:

Area

Wikipedia

Singular
area

Plural
areas or areæ

area (plural areas or areæ)

(mathematics) A measure of the extent of a planar region, or of the surface of a solid; it is measured in square units
A particular geographic region.
Any particular extent of surface.

The photo is a little dark in that area.
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.
(British) An open space, below ground level, between the front of a house and the pavement

Derived terms

Translations

measure of planar extent

Bulgarian: площ bg(bg), пространство bg(bg), повърхнина bg(bg)
Czech: plocha cs(cs) f.
Esperanto: areo eo(eo)

Japanese: 面積 ja(ja) (menseki)
Slovene: površina sl(sl) f.
Swedish: area sv(sv) c.

particular geographic region

Bulgarian: област bg(bg), район bg(bg)
Czech: oblast cs(cs) f.
Japanese: 領域 ja(ja) (ryōiki)

Scottish Gaelic: àrainn gd(gd) f.
Swedish: område sv(sv) n.

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.

Translations to be checked

Danish: areal n.
Dutch: gebied n., oppervlakte f.
Finnish: pinta-ala, alue
French: aire f.
Galician: área
German: Gebiet n., Areal
Hebrew: אזור (ezor) m.
Interlingua: area

Italian: area f.
Latin: regio f.
Polish: obszar m.
Portuguese: área f.
Russian: площадь f.
Spanish: área

Anagrams

Anagrams of aaer
æra

Galician

Noun

area f. (plural areas)

sand (grain)
sand (collectively)

Synonyms

(sand collectively): xabre

Italian

Noun

area f. (plural aree)

area
land, ground

Anagrams

Anagrams of aaer
aera

Latin

Noun

ārea (genitive āreae); f, first declension

open space
a threshing floor
vocative singular of ārea

āreā f.

ablative singular of ārea

Inflection

First declension (1).

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

Papiamentu

Noun

area

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.

(geometry) area; a measure of squared distance.

Hidden category: Check translations

Genealogy

Up to date as of February 01, 2010

From Familypedia

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.

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From Wikipedia, the free encyclopedia

Contents

Units

Formulae

Additional formulae

Areas of 2-dimensional figures

Area in calculus

Surface area of 3-dimensional figures

General formula

Area minimisation

See also

References

Notes

External links

From LoveToKnow 1911

Definition from Wiktionary, a free dictionary

Contents

English

Pronunciation

Etymology

Noun

Derived terms

Translations

See also

Anagrams

Galician

Noun

Synonyms

See also

Italian

Noun

Anagrams

Latin

Noun

Inflection

Papiamentu

Noun

Swedish

Noun

From Familypedia

Units