Derivation of the Formula for the Frustum

Date: 08/09/99 at 17:25:29 
From: Rizza
Subject: Frustum of a Right Circular Cone

I am trying to prove that the volume of a frustum of a right circular
cone, which is pi*(r^2 + rR = R^2)/3, is equal to the volume of the
solid (frustum of a cone) below the plane, where the base of the cone
is of radius r and height H is cut by a plane parallel to and h units
above the base. My problem is trying to prove that the answer is equal
to the equation for the volume of the frustum of a right circular cone
given above. 

Date: 08/09/99 at 22:37:07 
From: Doctor Jaffee 
Subject: Re: Frustum of a Right Circular Cone

I think I can help you out. First of all, your formula for the volume
of a frustum is not exactly correct. You have an equals sign in the
expression which I think is a typographical error. I assume you meant
to have an addition sign there. But more importantly, you left the H
out of the formula which should have been multiplied by pi. So, the
correct formula for the frustum of a right circular cone is

            pi*H(r^2 + rR + R^2)/3

Now, suppose that you have a cone with a base of radius r and height H
and it is cut by a plane parallel to and h units above the base. You
want to prove that the volume is

            pi*h(x^2 + rx + r^2)/3,

where x is the radius of the circle where the cone was cut by the
parallel plane.

Let A be the center of the circle at the base of the cone, B the
center of the circle up where the plane sliced the cone, and C the
vertex of the cone. Let E be a point on the circle at the base of the
cone, and D the point on the segment CE which intersects the circle
with center B.

You should then see that triangle AEC is similar to triangle BDC, so
corresponding sides are proportional.

Specifically,

            (H - h)/H = x/r

Cross multiply and get

            rH - rh = xH

Add rh - xH to both sides and get

            rH - xH = rh

which is equivalent to H(r - x) = rh.

So, if you divide both sides by r - x, you get

            H = rh/(r - x).

Now, the volume of the frustum is the volume of the original cone
minus the volume of the cone that was removed.

Use r for the radius and rh(r - x) for the height of the original cone
and use x for the radius and rh(r - x) - h for the height of the cone
that was removed.

If you do the algebra correctly, you will end up with the desired
result.

Give it a try and write back if my explanation needs clarification or
if you have difficulties.

- Doctor Jaffee, The Math Forum
http://mathforum.org/dr.math/