It is really a Brilliant symbol

There are 5 different length diagonals which are numbered 1,2,3,4,5 in the figure.

Type 1 diagonals are only 6 in number while 2,3,4 and 5 type diagonals are 12 each. The angle they subtend at the centre are $\pi, \frac{5\pi}{6},\frac{2\pi}{3},\frac{\pi}{2},\frac{\pi}{3}$ respectively. Using cosine rule their lengths are 2Rsinx, where x is half of the angle these diagonals subtend at the centre. $\text{While the length of each side is 2Rsin}\frac{\pi}{12} \text{and subtend }\frac{\pi}{6} \text {at the centre.}$ Hence required sum= $12\times 2R(\sin\frac{\pi}{12}+ \sin\frac{5\pi}{12}+\sin\frac{\pi}{3}+sin \frac{\pi}{4}+ \sin\frac{\pi}{6})+6\times 2R\sin\frac{\pi}{2}$ $\color{#69047E}{=144(2+\sqrt{6}+\sqrt{3}+\sqrt{2})}$ $\color{#D61F06}{p+q+r+s=144\times 5=720}$