Every formula I've found online is based on only a part of the total edge. If anyone knows the formulas based on each red edge below I would greatly appreciate it. A derivation of those formulas would be even better.
2026-02-23 06:41:01.1771828861
What are the formulas for the circumradius, surface area and volume of each Kepler-Poinsot polyhderon based on the length of the entire edge?
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As I understood from the discussion in the comments, a great stellated dodecahedron with the pentagram chord $c$ has circumsphere radius $$r_c=c\cdot \frac{\sqrt{3}\cdot (3+\sqrt{5})}{4\cdot (2+\sqrt{5})}= c\cdot \frac{\sqrt{3}\cdot (\sqrt{5}-1)}4,$$ surface area $$A=c^2\cdot \frac{15\sqrt{5+2\sqrt{5}}}{(2+\sqrt{5})^2}=c^2\cdot \frac{15\sqrt{5+2\sqrt{5}}}{9+4\sqrt{5}},$$ and volume $$V =c^3\cdot \frac{5\cdot (3+\sqrt{5})}{4\cdot (2+\sqrt{5})^3}=c^3\cdot \frac{5\cdot (13\sqrt{5}-29)}{4}.$$
To avoid the ambiguity with the multiplication and the division in the formulae at the linked page, I used the provided calculator to numerically check the final formulae above, plugging into them $c=1$.