Polygons
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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.
\(N\) | 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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