Question

In the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be StartFraction pi Over 4 EndFraction times the volume of the pyramid that it fits inside.

A cone is inside of a pyramid with a square base. The cone has a radius of r and a height of h. The length of the base edge of the pyramid is 2 r.


Which expression represents the volume of the cone that is StartFraction pi Over 4 EndFraction times the volume of the pyramid that it fits inside?

StartFraction pi Over 4 EndFraction(2r2h)
StartFraction pi Over 4 EndFraction(4r2h)
StartFraction pi Over 4 EndFraction(StartFraction r squared h Over 3 EndFraction)
StartFraction pi Over 4 EndFraction(StartFraction 4 r squared h Over 3 EndFraction)

Answer 1

Volume of the pyramid = (1/3)·(base area)·h = (1/3)·(2r)^2·h = (1/3)·4r^2·h.

So the cone's volume = (π/4)·(pyramid volume) = (π/4)·(4r^2h/3), which simplifies to πr^2h/3.

Thus the correct choice is: StartFraction pi Over 4 EndFraction(StartFraction 4 r squared h Over 3 EndFraction).