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
|
(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 variously gives the signed area of a parametric curves specified
as with
and the region on left side as the curve is traversed
as
|
(8)
| |||
|
(9)
| |||
|
(10)
|
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.