Wild About Math! Making Math fun and accessible

The amazing volume formula

More Fun With Mathematics by Jerome Meyer is a nice little book of interesting Math explorations. It's out of print but Amazon has a few very inexpensive used copies available. In the book I discovered this very odd volume formula that I've never seen before and couldn't find via Google. The author calls the formula "The Amazing Prismoidal Formula."

The formula states the following for any regular solid:

V = H*(B+4M+T)/6

where:

V = volume
H = height
B = area of the base
M = area of the middle of the solid
T = area of the top of the solid

Take a cube with side = 2 as a simple example:

H = 2
B = 2^2 = 4
M = B = 4
T = M = B = 4

V = H*(B+4M+T)/6 = 2*(4+4*4+4)/6 = 8, which is 2^3.

Use a cone with a base of radius r as another example, ignoring for the moment that it's not a regular solid. We'll get back to that.

H is not fixed. It can be any value.
B = pi*r^2
M = (pi*r^2)/4 since the radius of the circle in the middle is 1/2 of the radius of the base
T = 0

V = H*(B+4M+T)/6 = H*((pi*r^2) + 4*(pi*r^2)/4 + 0 )) /6 = (H*pi*r^2)/3, which is the familiar formula for the volume of a cone.

Meyer claims this formula works for any regular solid. Well, I think of regular solids as the 5 platonic solids. Meyer has illustrations of a cube, cone, cylinder, sphere, and conic frustrum (truncated cone).

I tried his formula on a tetrahedron and it works. I could not get it to work for an octahedron but that might have been an algebra mistake on my part. As the number of sides of the regular polygon increases determining the height and the area of the middle becomes more difficult.

An interesting exploration would be to determine for what solids, regular or not, does this formula work.