# Volume of cone in terms of base

Let $$K$$ be a $$d$$-dimensional cone with base $$L$$ and perpendicular height $$t$$ (that is, $$t$$ is the distance from the apex of $$K$$ to the hyperplane of $$L$$). Then $$\operatorname{Vol}_d (K) = \frac{1}{d}t\operatorname{Vol}_{d-1} (L).$$

Does anyone have any references on this result? If not, then how does one think about why it's true?

If you integrate the volume over the height, you get $$\int_0^tA\frac{h^{d-1}}{t^{d-1}}dh$$ where A is the “base”. This is because the cross sectional area is proportional to the d-1th power of the scale factor of this cross section (when compared to the base). Doing this integral gives the required result.