A Third of a Cylinder

Here is a way to see the third. Fill a cone with water and pour it into a cylinder with the same base and height. After one cone the cylinder is a third full, after three it is full. The narrowing point costs two thirds of the space.

Here both solids have radius 2 and height 6. The cylinder holds 24π and the cone 8π, one third of it. The two formulas say the same thing: the cone's has an extra 1/3 in front.

You can use this either way. Find the cylinder's volume and divide by 3 for the cone, or find the cone's and multiply by 3 for the cylinder.