(related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere)
So, (it is) clear, from this, that the square on the diameter of the sphere is five times the square on the radius of the circle from which the icosahedron has been described, and that the diameter of the sphere is the sum of (the side) of the hexagon, and two of (the sides) of the decagon, inscribed in the same circle.
(not yet contributed)
If the radius of the sphere is unity then the radius of the circle is \[\frac 2{\sqrt{5}},\] and the sides of the hexagon, decagon, and pentagon / icosahedron are \[\frac 2{\sqrt{5}},\quad 1-\frac 1{\sqrt{5}},\quad\text{ and }\quad \frac {\sqrt{10-2\,\sqrt{5}}}{\sqrt{5}},\] respectively.
Proofs: 1
Proofs: 1