Equilateral pentagon

Convex equilateral pentagon dissected into 3 triangles, which helps to calculate the value of angle δ as a function of α and β.

When a convex 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 ${\displaystyle \beta }$.

According to the law of sines the length of the line dividing the green and blue triangles is:

${\displaystyle a=2\sin \left({\frac {\beta }{2}}\right).}$

The square of the length of the line dividing the orange and green triangles is:

${\displaystyle {\begin{aligned}b^{2}&=1+a^{2}-2(1)(a)\cos \left(\alpha -{\frac {\pi }{2}}+{\frac {\beta }{2}}\right)\\&=1+4\sin ^{2}\left({\frac {\beta }{2}}\right)-4\sin \left({\frac {\beta }{2}}\right)\sin \left(\alpha +{\frac {\beta }{2}}\right).\\\end{aligned}}}$

According to the law of cosines, the cosine of δ can be seen from the figure:

${\displaystyle \cos(\delta )={\frac {1^{2}+1^{2}-b^{2}}{2(1)(1)}}\ .}$

Simplifying, δ is obtained as function of α and β:

${\displaystyle \delta =\arccos \left[\cos(\alpha )+\cos(\beta )-\cos(\alpha +\beta )-{\frac {1}{2}}\right].}$

The remaining angles of the pentagon can be found geometrically: The remaining angles of the orange and blue triangles are readily found by noting that two angles of an isosceles triangle are equal while all three angles sum to 180°. Then ${\displaystyle \epsilon ,\gamma ,}$ and the two remaining angles of the green triangle can be found from four equations stating that the sum of the angles of the pentagon is 540°, the sum of the angles of the green triangle is 180°, the angle ${\displaystyle \gamma }$ is the sum of its three components, and the angle ${\displaystyle \epsilon }$ 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.[1]

Cairo pentagonal tiling by equilateral pentagons with two non-adjacent right angles

There are two infinite families of equilateral convex pentagons that tile the plane, one having two adjacent complementary angles and the other having two non-adjacent complementary angles. Some of these pentagons can tile in more than one way, and there is a sporadic example of an equilateral pentagon that can tile the plane but does not belong to either of these two families; its angles are 89°16', 144°32'30", 70°55', 135°22', and 99°54'30", none complementary.[2]

All the equilateral pentagons plotted within the area delimited by the condition α ≥ β ≥ δ. Three regions for each of three types of pentagons are shown: stellated, concave and convex

Equilateral pentagons can intersect themselves either not at all, once, twice, or five times. The ones that don't intersect themselves are called simple, and they can be classified as either convex or concave. We here use the term "stellated" to refer to the ones that intersect themselves either twice or five times. We rule out, in this section, the equilateral pentagons that intersect themselves precisely once.

Given that we rule out the pentagons that intersect themselves once, we can plot the rest as a function of two variables 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 and the condition α ≥ β ≥ δ permit the size of the mapping to be limited. In the plane with coordinate axes α and β, α = β is a line dividing the plane in two parts (south border shown in orange in the drawing). δ = β as a curve divides the plane into different sections (north border shown in blue).

Both borders enclose a continuous region of the plane whose points map to unique equilateral pentagons. Points outside the region just map to repeated pentagons—that is, pentagons that when rotated or reflected can match others already described. Pentagons that map exactly onto those borders have a line of symmetry.

Inside the region of unique mappings there are three types of pentagons: stellated, concave and convex, separated by new borders.