The study of circumsphere and insphere of a regular polyhedron

Abstract

A regular polygon is a polygon that is both equilateral and equiangular. One of the properties of the regular polygon is has a circumcircle and an incircle. The analogy of a regular polygon on plane is a regular polyhedron in space, while the analogy of a circle on plane is a spehe in space. Relationship between regular polygon and circle on plane can be studied in space. The purpose of this paper is to show the existance and the characteristics of a circumsphere and an insphere of the regular polyhedron. The result show that each regular polyhedron has a circumsphere and an insphere. The center of the circumsphere of a regular polyhedron is intersection of the perpendicular bisector planes of the regular polyhedron and the center of the insphere of a regular polyhedron is intersection of the angle bisector planes of the regular polyhedron. The characteristics of a circumsphere and an insphere of a regular polyhedron are: 1) The center of a circumsphere and an insphere of a regular polyhedron is coincide; 2) The length of radius of a circumsphere and an insphere of a regular polyhedron meet the equation .