How do i prove the formula for the volume of a cone?

Question

I need a general proof for any form of a cone, not a right circular one.

Answer 1

Suppose that the base has area $a$ and the cone has height $h$, measured perpendicular to the base. Choose a system of coordinates, so that the cone point is at $z = 0$, and the altitude meets the base at $z = h$. In general, the cross section at arbitrary $z$ is a figure similar to the base with area $$ A(z) = a \left( \frac{z}{h} \right)^2. $$

Now integrate to find the volume. $$ \begin{align} V &= \int_0^h A(z) \, dz \\ &= \int_0^h a \frac{z^2}{h^2} \, dz \\ &= \left. \frac{a}{h^2} \frac{z^3}{3} \right|_0^h \\ &= \frac{ah}{3}. \end{align} $$