Equilateral pentagon

In geometry an equilateral pentagon is a polygon with five sides of equal length. Its five internal angles, in turn, can take a range of sets of values, thus permitting it to form a family of pentagons. The requirement is that all angles must add up to 540 degrees and must be between 0 and 360 degrees but not equal to 180 degrees. In contrast, the regular pentagon is unique, because it is equilateral and moreover it is equiangular (its five angles are equal; the measure is 108 degrees).

Four intersecting equal circles arranged in a closed chain are sufficient to determine a convex equilateral pentagon. Each circle's center is one of four vertices of the pentagon. The remaining vertex is determined by one of the intersection points of the first and the last circle of the chain.

It is possible to describe any convex equilateral pentagon with only two angles α and β with α ≥ β provided the fourth angle (δ) is the smallest of the rest of the angles. Thus the general equilateral pentagon can be regarded as a bivariate function f(α, β) where the rest of the angles can be obtained by using trigonometric relations. The equilateral pentagon described in this manner will be unique up to a rotation in the plane.

Regular pentagon

Regular star
pentagram

Convex

Adjacent right angles

Concave

Degenerate
(edge-vertex overlap)

Degenerate into triangle
(colinear edges)

Degenerate into trapezoid
(colinear edges)

Self-intersecting

When the equilateral pentagon is dissected into triangles, two of them appear as isosceles (triangles in orange and blue) while the other one is more general (triangle in green). We assume that we are given the adjacent angles ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.

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