Platonic Solid Inradii

Let $n$ be the number of faces, $A$ be the area of one face, $V$ be the volume, and $r$ be the inradius of one of the Platonic solids.

Then the Platonic solid can be subdivided into $n$ pyramids, where the apex of each pyramid is the center of the Platonic solid and the base of each pyramid is a face of the Platonic solid. Since each pyramid has a volume of $\frac{1}{3}Ar$ , the volume of the whole Platonic solid can be expressed as $V = \frac{1}{3}nAr$ .

Since the Platonic solid has an equal numerical surface area and volume, $V = S$ , and since the surface area is $S = nA$ , and $V = nA$ .

Therefore, $V = \frac{1}{3}nAr$ becomes $V = \frac{1}{3}Vr$ , which simplifies to $r = 3$ .

This means that all $5$ different Platonic solids with an equal numerical surface area and volume have an inradius of $3$ , which means that the sum is $5 \cdot 3 = \boxed{15}$ .