radiusin

Radiusin is a geometric term used to denote the ratio of a figure’s inradius to its circumradius. The inradius, r, is the radius of the largest circle contained entirely within the figure, while the circumradius, R, is the radius of the smallest circle that contains the figure. The radiusin of a shape is defined as radiusin = r/R, provided the shape has both an incircle and a circumcircle. This condition is met by many regular polygons and by certain polyhedra that are both tangential and cyclic.

For a regular n-gon, the radiusin equals cos(pi/n). This yields r/R values such as 1/2 for an equilateral triangle, cos(pi/4) ≈ 0.707 for a square, and cos(pi/5) ≈ 0.809 for a regular pentagon. In three dimensions, radiusin can be defined for convex solids that admit both an insphere (tangent to all faces) and a circumsphere (touching all vertices). In those cases, radiusin = r/R, and the value approaches 1 for shapes that are nearly spherical (for example, a sphere itself has radiusin = 1). For a cube, radiusin is 1/√3 ≈ 0.577.

Applications of the radiusin concept include using it as a simple measure of roundness or compactness in geometry, computer graphics, and mesh quality assessment. Limitations arise because not all polygons or polyhedra possess both an incircle and a circumsphere; thus the ratio is not universally defined. See also inradius, circumradius, and related roundness metrics.