Definition

The volume of a cone with base radius $r$ and perpendicular height $h$ is $V = (1/3)\pi r^2 h$, exactly one-third of the cylinder $V = \pi r^2 h$ with the same base and height, since the cross-sectional area decreases linearly from base to apex. Doubling only the height doubles the volume, while doubling only the radius quadruples it. As an integral, $$V = \int_0^h \pi \left(\frac{rz}{h}\right)^2 dz = \frac{\pi r^2}{h^2} \cdot \frac{h^3}{3} = \frac{1}{3}\pi r^2 h,$$ where the cross-section at height $z$ is a disk of radius $r(z/h)$, scaling linearly from $0$ at the apex to $r$ at the base; the same formula holds for an oblique cone, with $h$ the perpendicular height from apex to base plane, by Cavalieri's principle.