6.55. In a triangular pyramid, the dihedral angles at the base edges are equal to $\alpha$. Find its volume if the lengths of the base edges are $a, b$ and $c$.
6.55. Let $h$ be the height of the pyramid, $V$ its volume, and $S$ the area of the base. According to problem 6.54, $h = r \tan \alpha$, where $r$ is the radius of the inscribed circle of the base. Therefore, $V = Sh / 3 = Sr \tan \alpha / 3 = S^2 \tan \alpha / 3p = (p-a)(p-b)(p-c) \tan \alpha / 3$, where $p \Rightarrow = (a+b+c) / 2$.