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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.

The area of a region can be computed in the Wolfram Language using Area [reg].

A triangle area is given by

A_Delta=1/2lh,
(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_(rectangle)=ab,
(3)

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

A_(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-gon)=1/4ns^2cot(pi/n).
(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

The area of a polar curve with equation r=r(theta) is

A=1/2intr^2dtheta.
(7)

In Cartesian coordinates, Green's theorem variously gives the signed area of a parametric curves specified as (x(t),y(t)) with t in [t_0,t_1] and the region on left side as the curve is traversed as

Since these formulas give 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 when applying the above formulas to a given self-intersecting curve.

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

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