Area

The area of a surface or lamina is the amount of material needed to "cover" it completely. The area of a surface or collection of surfaces bounding a solid is called, not surprisingly, the surface area.

A triangle area 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)

In Cartesian coordinates, Green's theorem (modified so that the region is on the right as increases) gives the signed area as

			(8)
			(9)

Since this formula gives the signed area, the areas of curves with self-intersections, such as the fish curve, must be computed as a sum of absolute values of the areas of their components. Note also that it is incorrect to simply take the absolute value of the integrand.

The generalization of area to three dimensions is called volume, and to higher dimensions is called content.

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