Randolph also has plans to construct a glass viewing room inside the pool. The room will sit even with the base of the pool with the pool’s water above it, allowing visitors to remain completely dry while enjoying refreshments. He wants this room to be in the shape of a right prism with a square base, as illustrated.

The right square pyramid inside a cylinder has the same height h.

Randolph wants the diagonal of the pyramid’s base to be equal to the diameter, or twice the radius, of the pool’s base. He also wants the pyramid’s height to be equal to the height of the pool.

In the space provided, calculate the volume of the right square pyramid in Randolph’s pool in terms of the radius of the pool’s base,. When making your calculations, recall that each side of the prism’s square base is of length .