Regular polyhedron a hemisphere, and vertices around a great circle. It is "regular" if the vertices are equally spaced. The hosohedron {2,"n"} is dual to the dihedron {"n",2}. Note that when "n" = 2, we obtain the polyhedron {2,2}, which is both a hosohedron and a dihedron. All of these have Euler characteristic 2. Regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that faces are congruent regular polygons
Regular polygon "n"-gon, the sum of the perpendicular distances from any interior point to the "n" sides is "n" times the apothem (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the "n"=3 case. The circumradius "R" from the center of a regular polygon to one of the vertices is related to the side length "s" or to the apothem "a" by For constructible polygons, algebraic expressions for these relationships exist; see Bicentric polygon#Regular polygons. The sum of the perpendiculars from a regular "n"-gon's vertices to any line tangent to the circumcircle