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.
**

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 s^{n-1}.

The n-dimensional volume of the cone, V_{n}(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.,

which by a change of variable to s=(1-y/H) becomes

V

= BH[(1/n)s

The general formula is then:

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) = πr^{2}. 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, 4πr^{2}.
Thus the volume of the ball of radius r is equal to (1/3)r(πr^{2}); i.e.,
(4/3)πr^{3}.

Generalizing, this means that

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.