The volume of a cone

Assume that the volume of a right circular cone is a constant fraction k times the volume of its circumscribing cylinder. Can we find k and hence the formula for volume of cone?

Let be the volume of a cone of radius r and height (n + 1) h. Then:

Let be the volume of a similar cone, with part of the above cone, of height nh.

By similar triangles, the radius of this cone should have radius:

.

Therefore:

The difference is a frustum of volume V, where:

Now, as n ®¥, V tends to the volume of a cylinder of radius r and height h. Therefore . (formula for cylinder)

But by (*) above , as n ®¥,

Therefore , and .

Finally, the volume of a cone is given by: