The Geometry of Fit: How Shapes Relate to Their Circles

Regular polygons have been known for a long time to be able to be inscribed in a circle, where all the polygon’s vertices lie on the circle’s circumference. A circle is an infinite set of points that are all equidistant from a central point, and a regular polygon will share as many points with that circle as it has sides. It is also known that one can construct a regular tangential polygon by having each of its sides be tangent to the circle. While these relationships are well-established, the specific properties and comparisons of their respective radii, particularly when looking at a regular tangential polygon and a regular circumscribed triangle, offer a rich area for study. This paper compares the radii produced by these two distinct geometric configurations, analyzing how the number of sides, n, and the nature of the polygon’s relationship to the circle—whether it is tangent to the interior (inscribed) or its sides are tangent to the exterior, affect its radius.Through a generalized formula for regular polygons with n sides, we mathematically demonstrate the relationship between the in-radius and circumradius and show how this relationship varies with n