Here is the problem:

Prove that the volume of a cone, (defined as the set of points on line segments joining a vertex, v, with a set of points in the same plane called the base) with height H, and base area B is .

What makes this tricky is at first it seems, with so many possible shapes for the base, there is no good way to express all of these 'different' areas with the same A(x). But, since the slices are similar we need only concern ourselves with the ratio of any side, radius, diagonal or width of a given slice to that same side, diagonal radius, or width on the base. With this we can find the area, independent of any specific formula (such as or )

To do this use the fact that the ratio of areas of similar plane figures is equal to the ratio of any one of their dimensions squared. And if we are looking for the area slice at x that ratio will be the same as the ratio of x to H, the total height.

So, this means that . So, now you have a formula for A(x).