Find the volume of the pyramid given dihedral angles and the side lengths of the base triangle

Question

The trianglural pyramid's base has sides 5, 6 and 7. All of the sides of the pyramid make a 60 degree angle with the base. Find the volume of the pyramid.

This was suggested to me by my student, but I am not even sure where to begin. Any hints would be appreciated.

Answer 1

The perpendicular from the top $D$ to the base plane $ABC$, is a point $P$. In the right angle triangle $\triangle DPA$, the angle between $PA$ and $DA$ is $60^\circ$, so $$PA=\frac{DP}{\tan 60^\circ}$$ Similarly, $PB=PC=PA$. Therefore $P$ is the circumcenter. Find it, calculate the height $PD$ from the above formula, and then you get the volume.

Answer 2

In the Andrei's notation we obtain: $$PA=\frac{5\cdot6\cdot7}{4S_{\Delta ABC}}=\frac{105}{2\sqrt{9\cdot4\cdot3\cdot2}}=\frac{35}{4\sqrt{6}}.$$ Now, $$DP=\frac{35}{4\sqrt6}\cdot\sqrt3=\frac{35}{4\sqrt2}$$ and $$V_{ABCD}=\frac{1}{3}\cdot6\sqrt6\cdot\frac{35}{4\sqrt2}=\frac{35}{2}\sqrt3.$$