Regular polytope impossible ("not constructible"). The simplest few regular polygons that are impossible to construct are the "n"-sided polygons with "n" equal to 7, 9, 11, 13, 14, 18, 19, 21... Constructibility in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical. Euclid's "Elements" gave what amount to ruler-and-compass constructions for the five Platonic solids. (See, for example, Euclid's Elements.) However, the merely practical question of how one might draw a straight line in space, even with a ruler, might lead
Frustum volume of a pyramidal frustum whose bases are "n"-sided regular polygons is where "a" and "a" are the sides of the two bases. For a right circular conical frustum and where "r" and "r" are the base and top radii respectively, and "s" is the slant height of the frustum. The surface area of a right frustum whose bases are similar regular "n"-sided polygons is where "a" and "a" are the sides of the two bases. Frustum In geometry, a frustum (plural: "frusta" or "frustums") is the portion of a solid (normally a cone or pyramid) that lies between one