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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,
\end{displaymath} (1)

where $l$ is the base length and $h$ is the height, or by Heron's Formula
\end{displaymath} (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,
\end{displaymath} (3)

where the sides are length $a$ and $b$. This gives the special case of
A_{\rm square}=a^2
\end{displaymath} (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).
\end{displaymath} (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.
\end{displaymath} (6)

The Area of a Polar curve with equation $r=r(\theta)$ is
A={\textstyle{1\over 2}}\int r^2\,d\theta.
\end{displaymath} (7)

Written in Cartesian Coordinates, this becomes
$\displaystyle A$ $\textstyle =$ $\displaystyle {1\over 2}\int\left({x{dy\over dt}-y{dx\over dt}}\right)\,dt$ (8)
  $\textstyle =$ $\displaystyle {1\over 2}\int(x\,dy-y\,dx).$ (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


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.

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© 1996-9 Eric W. Weisstein