Thayer Watkins
Silicon Valley
& Tornado Alley

Area-Volume Formulas
for N-Dimensional Geometric Figures

There are sequences of geometric figures, such as a line, a square and a cube, which are dimensional analogues. Another sequence of dimensional analogues is that of the circle and the sphere. The terms circle and sphere are often used for what more properly should be called a disk and a ball. The perimeter of a disk is a circle and the surface of a ball is a sphere. Triangles and pyramids are two and three dimensional analogues of one another. The formulas for the length-area-volume of a line, a square and a cube whose sides are of length s are s, s2, s3. Obviously the volume of an n-dimensional cube of side s is sn. The area-volume formula for a disk and ball of radius r are πr2 and (4/3)*πr3 and it is not obvious what the general formula for the volume of an n-dimensional ball is. The perimeter of a disk of radius r is 2πr and the surface of a ball of radius r is 4πr2 and again the general rule is not obvious.

For triangles and pyramids the formulas are

Area = (1/2)Height*Base
Volume = (1/3)Height*Base

These relations can be used to express the area of a disk and the volume of a ball as

Area of disk = (1/2)Radius*Circumference
Volume of ball = (1/3)Radius*(Surface Area)

These, of course, reduce to the familiar formulas

Area of disk = (1/2)r*(2πr) = πr2
Volume of ball = (1/3)r*(4πr2) = (4/3)πr3

The previous formulas for triangles and pyramids suggest the general rule for an n-dimensional pyramid is that its volume is equal to (1/n)Height*(Base area).

This relation and the general cases for pyramids, cones, balls and ellipsoids are considered in the following sites.

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