Viviani's theorem for the n = 3 case. Circumradius 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 equals n times the circumradius. The sum of the squared distances from the vertices of a regular n-gon to any point on its circumcircle equals 2nR2 where R is the circumradius. The sum of the squared distances from the midpoints of the sides of a regular n-gon to any point on the circumcircle is 2nR2 − ns2, where s is the side length and R is the circumradius. If are the distances from the vertices of a regular -gon to any point on its circumcircle, then . Dissections Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into or parallelograms. These tilings are contained as subsets of vertices, edges and faces in orthogonal projections m-cubes. In particular, this is true for any regular polygon with an even number of sides, in which case the parallelograms are all rhombi. The list gives the number of solutions for smaller polygons. Area The area A of a convex regular n-sided polygon having side s, circumradius R, apothem a, and perimeter p is given by For regular polygons with side s = 1, circumradius R = 1, or apothem a = 1, this produces the following table: (Note that since as , the area when tends to as grows large.) Of all n-gons with a given perimeter, the one with the largest area is regular. Constructible polygon Some regular polygons are easy to construct with compass and straightedge; other regular polygons are not constructible at all. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon. This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later,