The volume of a cone is one-third the volume of a cylinder with the same base and height. This can be understood through a simple geometric explanation.
A cylinder and a cone share the same base radius and height. The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius of the base and h is the height.
For a cone, the volume is calculated using the formula V = (1/3)πr²h. The factor of 1/3 comes from the way a cone is formed. Imagine slicing a cylinder into three equal parts. Each part would be a cone with the same base and height as the original cylinder. Therefore, the volume of one cone is one-third of the cylinder’s volume.
This relationship can also be demonstrated using calculus by integrating the area of circular slices of the cone from the base to the apex. The result will always show that the volume of the cone is one-third that of the cylinder.
In summary, the volume of a cone is 1/3 of a cylinder because the cone occupies one-third of the space that a cylinder with the same base and height would occupy.