Area

The Area of a Surface is the amount of material needed to ``cover'' it completely. The Area of a Triangle is given by

$$A_\Delta ={\textstyle{1\over 2}}lh,$$ (1)

where $l$ is the base length and $h$ is the height, or by Heron's Formula

$$A_\Delta=\sqrt{s(s-a)(s-b)(s-c)},$$ (2)

where the side lengths are $a$, $b$, and $c$ and $s$ the Semiperimeter. The Area of a Rectangle is given by

$$A_{\rm rectangle} =ab,$$ (3)

where the sides are length $a$ and $b$. This gives the special case of

$$A_{\rm square}=a^2$$ (4)

for the Square. The Area of a regular Polygon with $n$ sides and side length $s$ is given by

$$A_{n-{\rm gon}}={\textstyle{1\over 4}}ns^2\cot\left({\pi\over n}\right).$$ (5)

Calculus and, in particular, the Integral, are powerful tools for computing the Area between a curve $f(x)$ and the x-Axis over an Interval $[a,b]$, giving

$$A=\int_a^b f(x)\,dx.$$ (6)

The Area of a Polar curve with equation $r=r(\theta)$ is

$$A={\textstyle{1\over 2}}\int r^2\,d\theta.$$ (7)

Written in Cartesian Coordinates, this becomes

$$A = {1\over 2}\int\left({x{dy\over dt}-y{dx\over dt}}\right)\,dt$$ (8)

$$= {1\over 2}\int(x\,dy-y\,dx).$$ (9)

For the Area of special surfaces or regions, see the entry for that region. The generalization of Area to 3-D is called Volume, and to higher Dimensions is called Content.

See also Arc Length, Area Element, Content, Surface Area, Volume

References

Gray, A. ``The Intuitive Idea of Area on a Surface.'' §13.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 259-260, 1993.