Simplex volume in terms of the radius of the inscribed sphere?

Question

It is well known that for the area of a triangle $A$ we have $$ A=r\cdot s,$$ where $s$ is the semiperimeter, and $r$ is the radius of the inscribed circle.

Is there an analogue for the higher-dimensional case. In other words, can I express the volume of a $d$-simplex in terms of the radius of its inscribed sphere and the volume of its boundary? If such a formula exists, what are the references?

Answer 1

The same holds, that in $\mathbb{R}^n$, $A = r \frac {V_b}{n}$, where $A$ is the volume of your simplex, and $V_b$ is the volume of your boundary.

To see why this works, simply show that a simplex with base of volume $B$ and height $r$ has volume $ r \frac {B}{n}$, and then add up over all faces.

Hint: The constant $\frac {1}{n}$ comes from $\int x^{n-1}\, dx = \frac {x^n}{n}$.