Integer triangle cosine equals 0, ±1/2, or ±1. The only ones of these giving an angle strictly between 0° and 180° are the cosine value 1/2 with the angle 60°, the cosine value –1/2 with the angle 120°, and the cosine value 0 with the angle 90°. The only combination of three of these, allowing multiple use of any of them and summing to 180°, is three 60° angles. Conditions are known in terms of elliptic curves for an integer triangle to have an integer ratio "N" of the circumradius to the inradius. The smallest case, that of the equilateral triangle, has
Equilateral pentagon formula_8 is the sum of its three components, and the angle formula_9 is the sum of its two components. A cyclic pentagon is equiangular if and only if it has equal sides and thus is regular. Likewise, a tangential pentagon is equilateral if and only if it has equal angles and thus is regular. The equilateral pentagon as a function of two variables can be plotted in the two-dimensional plane. Each pair of values (α, β) maps to a single point of the plane and also maps to a single pentagon. The periodicity of the values of α and β