## Area

The Area of a Surface is the amount of material needed to cover'' it completely. The Area of a Triangle is given by (1)

where is the base length and is the height, or by Heron's Formula (2)

where the side lengths are , , and and the Semiperimeter. The Area of a Rectangle is given by (3)

where the sides are length and . This gives the special case of (4)

for the Square. The Area of a regular Polygon with sides and side length is given by (5)

Calculus and, in particular, the Integral, are powerful tools for computing the Area between a curve and the x-Axis over an Interval , giving (6)

The Area of a Polar curve with equation is (7)

Written in Cartesian Coordinates, this becomes   (8)  (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.

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.