In the geometry lingo, this $\pi$e is what is called the circle inscribed to the triangle, and the way to measure its radius – which is the essential parameter from which we will derive its area – is by leveraging precisely this condition.
For the inexperienced reader, pause and try to convince yourself that such a circle intersects with its bounding shape forming ${90}^{\circ }$-angles on such points; using this result we will now divide the original box into three smaller ones, which share the property that their height is the radius we are looking for: With this key insight in mind, we can now relate the area of the big triangle, to that of the smaller ones, and thus to $r$; that is, $$[△ABC]=\frac{1}{2}13r+\frac{1}{2}14r+\frac{1}{2}15r=21r⟹r=\frac{[△ABC]}{21}.$$
However, we still have not figured out $[△ABC]$, and it is Heron's formula the one that tells us exactly this. Letting $s$ be half of the perimeter, then $$[△ABC]=\sqrt{s(s-13)(s-14)(s-15)}=\sqrt{21·8·7·6}=84,$$ so finally, $$r=\frac{84}{21}=4⟹[\pi ]=\pi ·4^2=16\pi .$$