This article shows the derivations of the many formulas that
are needed to work with the polygons and polyhedra that are of interest to
woodturners interested in turning spheres and items derived from spheres. In a companion article^{[1]} the formulas
are given without any algebraic and geometric details. This was done in the knowledge that most
woodturners will have little to no interest in the details. However this article is directed at the
small few who are interested in that detail.

In the sections dealing with regular polygons, the focus is
on finding the radius of the circumscribed circle and the radius of the
inscribed circle. The **circumscribed
circle** is the circle that touches the polygon at all of its vertices and at
no other point. The **inscribed circle**
is the circle that touches the polygon at the midpoint of each side and at no
other point.

In the sections dealing with regular polyhedra, the focus is on finding the radius of the circumscribed sphere and the radius of the inscribed sphere. The **circumscribed sphere ** is the sphere that touches the polyhedron at all of its vertices and at no other point. The **inscribed sphere ** is the sphere that touches the polyhedron at the centroid of each face and at no other point.