# What is the ratio of the volume of the pyramid to the volume of the cube?

Problem

A sphere is inscribed inside a pyramid with a square as a base whose height is $15$ times the length of one edge of the base. A cube is inscribed inside the sphere. What is the ratio of the volume of the pyramid to the volume of the cube?

Official solution

Question The problem never mentions the side lengths of the pyramid, only the height. Therefore how can the solution conclude that the plane cut is isosceles?

• Is it because the pyramid is inscribable with a sphere? That leads to the question of where it is possible to inscribe a sphere inside a square pyramid that isn't equilateral? – John Ryan Dec 23 '15 at 19:21