We can work backward from a given volume to find possible dimensions for a cone or pyramid.

Suppose we want to find dimensions for a cone so it has a volume of cubic inches. Start by substituting the volume into the pyramid volume formula to get . The base of a cone is a circle, so we can write . Multiply both sides of the equation by 3 and divide both sides by to get .

Now consider different possible values for and . If we can find a perfect square that divides evenly into 2,700, we can set the square root of that number to be the radius. The number 25 is a perfect square and divides into 2,700, so choose . Now . This tells us that if the pyramid’s radius is 5 inches, its height is 108 inches because .

These aren’t the only possible values. Suppose we set the radius to be 20 inches. Substitute this into the original equation and rearrange to find the value of .

A height of 6.75 inches together with a radius of 20 inches gives the cone a volume of  cubic inches.