The height of the pyramid in the diagram is three times the radius of the cone. The base area of the pyramid is the same as the base area of the cone. What is the expression for the volume of the pyramid in terms of the radius r of the cone?

Since the height of the pyramid is 3 times the radius of the cone we can express height of the pyramid as a function of radius of the cone.

Let radius of cone = r
Then, height of the pyramid = 3r

Area of the base of the cone = pi x r^2 = area of the pyramid.

If the base dimensions of the pyramid are l and b, then area of the pyramid, l x b = pi x r^2.

The volume of the pyramid = 1/3 x l x b x h

We know, l x b = pi x r^2 and h = 3r.

Thus, volume of pyramid is = 1/3 x pi x r^2 x 3 r = pi x r^3