Mostly elementary geometry.
Simple family of pentagons. Each member can be described with a single positive real number r. The pentagons have four identical angles A, four identical sides s, a fifth angle B and a fifth side t. In all these pentagons:
- r = B / A
- B = 3π + 4A
- t = 2s(cos(2A) + cos(2A - π))
Both the regular pentagon {5} and the equilateral pentagram {5/2} belong to the family.
See AsAsBsAsAt.
All the equilateral pentagons.
A more complex family of pentagons. Each member can be described with two positive real numbers r1 and r2. The pentagons have the five sides equal s but can have the five angles A, B, C, D, E different.
See AsBsCsDsEs.
Simple family of heptagons. Each member can be described with a single positive real number r. The heptagons have six identical angles A, six identical sides s, a seventh angle B and a seventh side t. In all these heptagons
- r = B / A
- B = 5π + 6A
- t = 2s(cos(2A) + cos(2A - π) + cos(3A - 2π))
The regular heptagon and the two equilateral heptagrams {7/2} and {7/3} belong to this family.
See AsAsAsBsAsAsAt.