# Introduction

My background has qualified me to write this rather long article. My formal education is in mathematics and physics. I have had a long and prosperous carreer as a software engineer. And I am a woodturner. (And a woodcarver, too.)

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.

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