# Polygons

Corresponding Wikipedia article: Regular polygon

A circle can be divided into N equal parts. The chords, which subtend these parts, produce a regular N-sided polygon. This polygon approximates the circumscribed circle.

Parameters of an N-sided polygon inside the unit circle
$$N$$ Prime factors Edge length Semiperimeter Shape
$$2$$ $$2$$ $$2$$ Segment.
$$3$$ $$3$$ $$\sqrt{3}$$ $$2,60$$ Equilateral triangle.
$$4$$ $$2\times 2$$ $$\sqrt{2}$$ $$2,83$$ Square made from segments intersected at 360⁰/4=90⁰.
$$5$$ $$5$$ $$\sqrt{2-\phi}$$ $$2,94$$ Regular pentagon.
$$6$$ $$2\times 3$$ $$1$$ $$3$$ Regular hexagon. Intersection of two equilateral triangles at 360⁰/6=60⁰ or 360⁰/2=180⁰.
$$7$$ $$7$$ $$2\sin\frac{\pi}{7}$$ $$3,04$$ Regular heptagon.
$$8$$ $$2\times 2\times 2$$ $$\sqrt{2-\sqrt{2}}$$ $$3,06$$ Regular octagon. Intersection of two squares at 360⁰/8=45⁰.
$$9$$ $$3\times 3$$ $$2\sin\frac{\pi}{9}$$ $$3,08$$ Regular nonagon. Intersection of three equilateral triangles at 360⁰/9=40⁰.
$$10$$ $$2\times 5$$ $$\phi$$ $$3,09$$ Regular decagon. Intersection of two regular pentagons at 360⁰/10=36⁰ or 360⁰/2=180⁰.
$$11$$ $$11$$ $$2\sin\frac{\pi}{11}$$ $$3,10$$ Regular hendecagon.
$$12$$ $$2\times 2\times 3$$ $$\frac{\sqrt{3}-1}{\sqrt{2}}$$ $$3,11$$ Regular dodecagon. Intersection of two regular hexagons or three squares at 360⁰/12=30⁰.
$$\infty$$ $$\pi$$ Circle.
$$\phi$$ – golden ratio
$$2\sin\frac{\pi}{9}=\sqrt[3]{-\frac{\sqrt{3}}{2}+\frac{i}{2}}-\sqrt[3]{\frac{\sqrt{3}}{2}+\frac{i}{2}}$$