Polygons

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Corresponding Wikipedia article: Regular polygon

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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 \(2\sin\frac{\pi}{9}=\sqrt[3]{-\frac{\sqrt{3}}{2}+\frac{i}{2}}-\sqrt[3]{\frac{\sqrt{3}}{2}+\frac{i}{2}}\)

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