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
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(1)
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where is the base length and is the height,
or by Heron's formula
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(2)
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where the side lengths are , , and and the semiperimeter.
The area of a rectangle is given
by
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(3)
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where the sides are length and . This gives the
special case of
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(4)
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for the square. The area of a regular polygon with sides and side
length is given by
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(5)
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Calculus and, in particular, the integral, are powerful tools for computing
the area between a curve and the x-axis
over an interval , giving
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(6)
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The area of a polar curve with equation is
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(7)
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In Cartesian coordinates, Green's theorem (modified so
that the region is on the right as increases) gives
the signed area as
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.
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.
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