TOPICS

# 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

 (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.

Arc Length, Area Element, Area Moment of Inertia, Content, Geometric Centroid, Green's Theorem, Polygon Area, Surface Area, Triangle Area, Volume Explore this topic in the MathWorld classroom

## Explore with Wolfram|Alpha

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## References

Gray, A. "The Intuitive Idea of Area on a Surface." §15.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 351-353, 1997.

Area

## Cite this as:

Weisstein, Eric W. "Area." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Area.html