The standard $n$-simplex contains all points $\vec{x} \in \mathbb{R}^{n + 1}$ such that $0 \le x_i \le 1$ and $\vec{x} \cdot \vec{1} = 1$
The standard 2-simplex is an equilateral triangle with side length $\sqrt{2}$ and vertices at (1, 0, 0), (0, 1, 0), and (0, 0, 1). The area is $\sqrt{3}/2$.
The standard 1-simplex is a line with vertices at (1, 0) and (0, 1). The length is $\sqrt{2}$
What is the area of the standard $n$-simplex?
Is it $\sqrt{n + 1} / n!$ ?
Best Answer
From this SE-quest you have the height $h_D$ of the unit-sided regular simplex as $$h_D=\sqrt{\frac{D+1}{2D}}$$ Further you obviously have the dimensional recursion on the volume $V_D$ $$V_D=\frac1D\ V_{D-1}\ h_D$$ With the obvious recursion start of $V_1=1$ you thus get $$V_D=\prod_{d\leq D}\frac{h_d}d=\frac1{D!}\sqrt{\frac{D+1}{2^D}}$$ (The factor $2^D$ in the denominator surely could be omitted, if you deal with a simplex with sidelength of $\sqrt2$ units instead.)
--- rk