Why is the volume of a cone one third of a cylinder?

Think of a block of clay shaped like a cylinder. If you wanted to carve it into a perfect cone with the same base, you would have to shave off the outer edges as you move up towards the center point. While it might look like you are carving away half the clay, you are actually removing exactly two-thirds of it because the volume decreases rapidly in three dimensions as the radius shrinks. The water pouring experiment is a classic way to prove this in a classroom: pouring three equal cones of liquid perfectly fills one equal cylinder.

Historically, mathematicians like Archimedes used a method of exhaustion to prove this relationship, breaking the shape into infinitely thin flat disks. In modern math, we use Calculus. If you integrate the area of the circular cross-sections of a cone from its tip to its base, the squared radius term in the area formula ($\pi r^2$) integrates to include a $\frac{1}{3}$ fraction. Even without calculus, Cavalieri's Principle shows that because a cone's cross-sectional area shrinks quadratically as you go up, its total volume must be one-third of the bounding cylinder.

A common mistake is confusing the height of the cone ($h$) with the slant height ($l$). The formula $V = \frac{1}{3}\pi r^2 h$ strictly requires the perpendicular height: the straight line dropping from the very top tip of the cone straight down to the center of the circular base. The slant height is only used for calculating surface area, not volume.