User:Ajwm8103/sandbox

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For my regular polygon addition:

Area

The area of a star polygon can be expressed in terms of the overlapping area, with circumradius R

$A={\tfrac {1}{2}}nR^{2}\sin \left({\frac {2\pi m}{n}}\right)$

The area can also be expressed as a non-overlapping area

$A=R^{2}\left(n\cos ^{2}\left({\frac {\pi m}{n}}\right)\tan \left({\frac {\pi }{n}}(1-m)\right)+{\frac {n}{2}}\sin \left({\frac {2\pi m}{n}}\right)\right)$

Add this back with citation later.

Area of a Regular Polygon (convex and star)

I was doing some work on regular polygons and Schläfli notation and came across a pretty condensed expression for the area of a regular polygon. $A=R^{2}{\frac {p}{2}}\sin {\frac {2\pi q}{p}}$ Where p & q come from the Schläfli symbol {^p/_q} and R is the circumradius of the regular polygon. This counts some sections of the star polygons multiple times, as seen in this image from the density page:

The boundary of the regular enneagram {9/4} winds around its centre 4 times, so it has a density of 4.

The star polygon page also contains some useful information in the Interiors section.

Could this expression be useful on this page? There's an interesting derivation that comes with it, if that's at all relevant.

Ajwm8103 (talk) 19:12, 29 January 2021 (UTC)

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