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.