The goal: Make the face for a 10-sided dice or a heptagonal trapezohedron with a circumsphere, using as input one side of a face or the radius of the sphere.
Let us be on the same page:
Circumsphere: Exist a sphere that touches every point of the polyhedron.
Antiprism:
...a polyhedron composed of two parallel direct copies (not mirror images) of an k-sided polygon, connected by an alternating band of 2k triangles...
-Wikipedia
A shape is the dual polyhedron of another: A vertice of one is transformed into a face of the other, and viceversa; but the edges stay in their place.
Trapezohedron: The dual polyhedron of an antiprism. The 2k faces of an k-gonal trapezohedron are the same and are kites (also called deltoids).
1.- The faces, as described earlier, are kites; with a long side LS and a short side SS. What must be the angles for this kite to be used for my polyhedron? Indicate if you're answering for a pentagonal or heptagonal trapezoihedron
2.- Now let's think I have the lenght of one side of the deltiod, How do I calculate the other so the kite can be a face of a trapezohedron with a circumsphere? Use l10(SS) and s10(LS) to answer this question with functions for a dice D10 or l14(SS) and s14(LS) for a heptagonal one.
2.- Now, let's imagine I have a limited space for a cube box with side 2R, and I'd like my D10 or D14 to fit in there, so the trapezohedron's circumsphere's radius must be R, but I don't know the sides of the faces LS and SS, I only know R. How do I calculate the sides of the deltoid only knowing R? Use L10(R) and S10(R) for the 10-sided dice and L14(R) and S14(R) for the 14-sided polyhedron
I really hope you can help for the customized heptagonal trapezohedron D14. English isn't my first language, I hope I wrote everything correctly.
PD: Don't confuse l14() and s14() with L14() and S14()