Thayer Watkins

Area-Volume Formulas
for N-Dimensional Pyramids and Cones

An n dimensional pyramid or cone is a geometric figure consisting of an (n-1) dimensional base and a vertical axis such that the cross-section of the figure at any height y is a scaled down version of the base. The cross-section becomes zero at some height H. The point at which the cross-section is zero is called the vertex. The distinction between a pyramid and cone is that the base of a pyramid is a geometric figure with a finite number of sides whereas there are no such restrictions for the base of a cone (and thus a pyramid is a special case of a cone).

A two dimensional pyramid is just a triangle and a three dimensional pyramid is the standard type pyramid with a polygonal base and triangular sides composed of the sides of the base connected to the vertex. The area-volume formulas for these two cases are well known: i.e.

Area of Triangle = (1/2)Base*Height

Volume of Pyramid/Cone
= (1/3)Base*Height

In order to deal with the general n dimensional case it is necessary to derive the area of the triangle systematically. The area of a triangle can be found as the limit of a sequence of approximations in which the triangle is covered by a set of rectangles as shown in the diagrams below.

In the above construction the vertical axis of the triangle is divided into m equal intervals. The width of a rectangle used in the covering is the width of the triangle at that height. As the subdivision of the vertical axis of the disk becomes finer and finer the sum of the areas of the rectangles approaches a limit which is called the area of the triangle.

The process can be represented algebraically. For a pyramid/cone of height H the distance from the vertex is H-y where y is the distance from the base. Let s=(1-y/H) be the scale factor for a cross-section of the cone at a height y above the base. The area of a the (n-1)-dimensional cross-section is equal the area of the base multiplied by a factor of sn-1.

The n-dimensional volume of the cone, Vn(B,H), is approximated by the sum of the volumes of the prisms created by the subdivision of the vertical axis. The limit of that sum as the subdivision becomes finer and finer can be expressed as an integral; i.e.,

Vn(B,H) = ∫H0B(1-(y/H))n-1dy
which by a change of variable to s=(1-y/H) becomes
Vn(B,H) = BH∫10sn-1ds
= BH[(1/n)sn]10 = (1/n)BH

The general formula is then:

Volume of an N-Dimensional Pyramid/Cone = (1/N)Base*Height

The above general formula can be used to establish a relationship between the volume of an n-dimensional ball and the (n-1)-dimensional area which bounds it. Consider the approximation of the area of a disk of radius r by triangles as shown below:

Each of the triangles has a height of r so the sum of the areas of the triangles is equal to the height r times the sum of the bases. In the limit the sum of the bases is equal to the perimeter of the circle so the area of the disk is equal to (1/2)r(2πr) = πr2. Likewise the volume of a ball can be approximated by triangulating the spherical surface and creating pyramids whose verices are all at the center of the ball and whose bases are the triangles at the surface. The height of all these pyramids is radius of the ball r. Thus the volue is equal to one third of the height r times the sum of the base areas. In the limit the sum of the base areas is equal to the area of the sphere, 4πr2. Thus the volume of the ball of radius r is equal to (1/3)r(πr2); i.e., (4/3)πr3.

Generalizing, this means that

Volume of
N-Dimensional Ball of Radius R
= (1/N)R(Area of Surface of Ball)

Unfortunately this relation is of no practical help in finding the formula for the volume of an n-dimensional ball in that the formula for the area of the surface of an n-dimensional ball is more obscure that that of the volume. Nevertheless it is interesting to perceive an n-dimensional ball as being composed of n-dimensional pyramids.

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