Hi Keith,

Unfortunately I don't have an intuitive explanation for the fraction 1/3. The modern way to develop this formula is to use calculus and the 1/3 comes from an antiderivative.

On the Math Forum site there is a development of an expression to approximate the volume of a cone by slicing the cone into thin layers parallel to the base. Each layer is approximately a cylinder and adding the volumes of these cylinders you get approximately the volume of the cone. The 1/3 in the expression that results comes from finding a compact formula for

\[1^2 + 2^2 + 3^2 + \cdot \cdot \cdot + n^{2}\]

where $n$ is the number of layers.

I know this is not a very satisfactory response to your question but it is the best response I have.

Penny