Volume of a Cone or Pyramid

Volume of a Pyramid or Cone

The Goal: Prove that if C is a pyramid or cone with height H and base area B, then the volume of C = (1/3)BH.

The Proof: Part is based on the "Facts" below, which you can use. The rest will be left to the assignments.

Fact 0. Definitions.

In 3-space, if R is a region in a plane and V is a point not in the plane, the (bounded) cone with base R with vertex V is the set of all points that are on segments VA, where A is a point in R. We may denote this as C(R,V).
In the previous definition, if R is a polygon, then C is called a pyramid with base R and vertex V. So a pyramid is a special case of a cone.
Note: the cone concept is very general. If R is a line segment AB, then C(R,V) = triangle ABV. If R is a circle with center O and VO is perpendicular to the plane of R, then the cone is called a right circular cone. This one is the "ice cream cone".

Fact 1. Length, Volume, Area and Scaling

This we assume without a careful proof, mainly because we have never given a careful definition of area and volume. We will rely on the calculus background for basic properties. The key fact we will use is this dilation scaling property for lengths, areas and volumes.

If T is a similarity transformation from 3-space to 3-space with ratio of similitude k, the length of the image of a segment of length L is kL. The image of a region with area A is k²A. The volume of the image of a solid of volume V has volume k³D.

Step 2. Cones and dilations

Recall that D_V,k -- the dilation with center V and ratio k -- is the transformation that takes a point P and moves it to the point P' on line OP so that k = OP'/OP. Negative ratios are possible. If k is positive, then P' is on ray OP and if k is negative P' is on the opposite ray. If k is nonzero, then the dilation is a similarity transformation with ratio |k|, Note: This definition works fine in 3-space as well as the plane. In fact it works in any dimension.

An alternate definition of the cone with base R and vertex V is the set of points D_{V,k(A), where A is in R and k runs from 0 to 1.}

Then a dilation with vertex V with ratio between 0 and 1 maps the cone C into "subcones" similar to the original cone. The base of such a "subcone" is a "slice" parallel the base. This is the intersection of the cone with_{a plane parallel to the base plane.}_{We can see that each
such slice = D_V,k(R) for some fixed k.}

Fact 3. Cavalieri's Principle.

Take any line m in 3-space. Suppose S1 and S2 are two shapes in 3-space. For every plane M perpendicular to line m, we can measure the area of the intersection of S1 with M and the area of the intersection of S2 with M. Call these areas A1(M) and A2(M). Cavalieri's Principle states that if for all M, A1(M) = A2(M), the volume S1 = volume S2.

Important Note: We have seen this before for regions in two-space, where if the lengths of the intersections are equal, then the areas are equal. However, in 3-space, it is possible for two polyhedra (including cones) to have plane intersections with equal length but still have different surface areas. So this principle only works to give equality of volumes in 3-space and equality of areas in 2-space.

Fact 4. Volume and Partial Scaling

Suppose in space we pick a point O and (x,y,z) coordinates. For some constants, r, s, t, let T be the transformation T(x,y,z) = (rx, sy, tz). Then for any region S in 3-space, the volume of T(S) = |rst| volume of S.

This fact is not surprising. If we approximate the volume of S by little cubes with sides parallel to the coordinates axes, T transforms a cube of side length L into a rectangular solid with side lengths rL, sL, tL.