Area Designer

Friday, 27 January 2012

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Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (m²), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays a important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

Formal definition

An approach to defining what is meant by area is through axioms. For example, we may define area as a function ''a'' from a collection ''M'' of special kind of plane figures (termed measurable sets) to the set of real numbers which satisfies the following properties:

For all ''S'' in ''M'',

a(S) \ge 0

If ''S'' and ''T'' are in ''M'' then so are

S\cup T

and

S\cap T

and also,

a(S\cup T)=a(S)+a(T)-a(S\cap T)

If ''S'' and ''T'' are in ''M'' with

S\subset T

then ''T'' − ''S'' is in ''M'' and ''a''(''T'' − ''S'') = ''a''(''T'') − ''a''(''S'').

If a set ''S'' is in ''M'' and ''S'' is congruent to ''T'' then ''T'' is also in ''M'' and ''a''(''S'') = ''a''(''T'').

Every rectangle ''R'' is in ''M''. If the rectangle has length ''h'' and breadth ''k'' then ''a''(''R'') = ''hk''.

Let ''Q'' be a set enclosed between two step regions ''S'' and ''T''. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e.

S\subset Q\subset T

. If there is a unique number ''c'' such that

a(S)\le c\le a(T)

for all such step regions ''S'' and ''T'', then ''a''(''Q'') = ''c''.

It can be proved that such a area function actually exists. (See, for example, ''Elementary Geometry from an Advanced Standpoint'' by Edwin Moise.)

Units

Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measure in square metres (m²), square centimetres (cm²), square millimetres (mm²), square kilometres (km²), square feet (ft²), square yards (yd²), square miles (mi²), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.

The SI unit of area is the square metre, which is considered an SI derived unit.

Conversions

The conversion between two square units is the square of the conversion between the corresponding length units. For example, since :1 foot = 12 inches, the relationship between square feet and square inches is :1 square foot = 144 square inches, where 144 = 12² = 12 × 12. Similarly:

1 square kilometer = 1,000,000 square meters

1 square meter = 10,000 square centimetres = 1,000,000 square millimetres

1 square centimetre = 100 square millimetres

1 square yard = 9 square feet

1 square mile = 3,097,600 square yards = 27,878,400 square feet

In addition,

1 square inch = 6.4516 square centimetres

1 square foot = square metres 1 square yard = square metres 1 square mile = square kilometres

Other units

There are several other common units for area. The are was the original unit of area in the metric system, with

1 are = 100 square metres

Though the are has fallen out of use, the hectare is still commonly used to measure land:

1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres

Other uncommon metric units of area include the tetrad, the hectad, and the myriad.

The acre is also commonly used to measure land areas, where

1 acre = 4,840 square yards = 43,560 square feet.

An acre is approximately 40% of a hectare.

Basic area formulae

Rectangles

The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length and , the formula for the area is : (rectangle). That is, the area of the rectangle is the length multiplied by the width. As a special case, the area of a square with side length is given by the formula : (square).

The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before arithmetic, this formula can be used to define multiplication of real numbers.

Dissection formulae

Most other simple formulae for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must sum to the area of the original shape.

For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: : (parallelogram). thumb|right|180px|Two equal triangles.However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of the parallelogram: : $A = \frac{1}{2}bh$ (triangle). Similar arguments can be used to find area formulae for the trapezoid and the rhombus, as well as more complicated polygons.

Circles

The formula for the area of a circle is based on a similar method. Given a circle of radius , it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form and approximate parallelogram. The height of this parallelogram is , and the width is half the circumference of the circle, or . Thus, the total area of the circle is , or : : (circle). Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the approximate parallelograms is exactly , which is the area of the circle.

This argument is actually a simple application of the ideas of calculus. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus. Using modern methods, the area of a circle can be computed using a definite integral: : $A \;=\; \int_{-r}^r 2\sqrt{r^2 - x^2}\,dx \;=\; \pi r^2.$

Surface area

Most basic formulae for surface area can be obtained by cutting surfaces and flattening them out. For example, if the side surface of a cylinder (or any prism) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone, the side surface can be flattened out into a sector of a circle, and the resulting area computed.

The formula for the surface area of a sphere is more difficult: because the surface of a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work ''On the Sphere and Cylinder''. The formula is : (sphere). where is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus.

List of formulae

The above calculations show how to find the area of many common shapes.

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

Additional formulae

Areas of 2-dimensional figures

a triangle:

\tfrac12Bh

(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 = \tfrac12(a + b + c)

is half of its perimeter. If an angle and its two included sides are given, the area is

\tfrac12 a b \sin(C)

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

\tfrac12(x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3)

. 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)'', and ''(x₃,y₃)''. 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. This result is known as Pick's theorem.

Area in calculus

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\theta

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

:: ${1 \over 2} \oint_{t_0}^{t_1} \vec u \times \dot{\vec u} \, dt.$

Surface area of 3-dimensional figures

cube:

6s^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

\pi r^2 + \pi r l

where ''r'' is the radius and ''l'' is the slant height of the cone.

\pi r^2

is the base area while

\pi 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}\times\frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv.

Minimization

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

External links

Area formulas

Conversion cable diameter to circle cross-sectional area and vice versa

af:Oppervlakte als:Flächeninhalt ar:مساحة an:Aria arc:ܫܛܝܚܘܬܐ az:Sahə (ölçü parametri) zh-min-nan:Biān-chek be:Плошча be-x-old:Плошча br:Gorread bg:Площ ca:Àrea cv:Лаптăк ceb:Langyab cs:Obsah sn:Ruvanze cy:Arwynebedd da:Areal de:Flächeninhalt dv:އަކަމިން dsb:Wopśimjeśe płoni et:Pindala el:Εμβαδόν es:Área eo:Areo eu:Azalera fa:مساحت fo:Vídd fr:Aire (géométrie) gv:Eaghtyr gd:Farsaingeachd gl:Área gan:面積 gu:ક્ષેત્રફળ ko:넓이 haw:ʻAlea hi:क्षेत्रफल hsb:Wobsah přestrjenje hr:Površina io:Areo ilo:Kalawa id:Luas ia:Area os:Фæзуат is:Flatarmál it:Area he:שטח jv:Jembar ka:ფართობი kk:Алаң ku:Rûerd lo:ເນື້ອທີ່ la:Area (geometria) lv:Laukums lb:Fläch lt:Plotas li:Oppervlak ln:Etando hu:Terület (matematika) mk:Плоштина mg:Velarantany ml:വിസ്തീർണ്ണം mr:क्षेत्रफळ ms:Keluasan mwl:Ária nl:Oppervlakte ja:面積 no:Areal nn:Flatevidd oc:Aira mhr:Кумдык pfl:Fläsch km:ក្រលាផ្ទៃ nds:Flach pl:Pole powierzchni pt:Área ro:Arie qu:Hallka k'iti k'anchar ru:Площадь se:Viidodat sco:Aurie simple:Area sk:Plocha (útvar) cu:Пространиѥ sl:Površina so:Bed ckb:ڕووبەر sr:Површина sh:Površina (geometrija) fi:Pinta-ala sv:Area tl:Lawak ta:பரப்பளவு te:విస్తీర్ణము th:พื้นที่ tg:Масоҳат tr:Alan uk:Площа ur:رقبہ vi:Diện tích vls:Ippervlak war:Kahaluag wo:Yaatuwaay wuu:面积 yi:שטח yo:Ààlà zh-yue:面積 zh:面积

This text is licensed under the Creative Commons CC-BY-SA License. This text was originally published on Wikipedia and was developed by the Wikipedia community.

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