# Polygons

## Regular Polygons
  * polygons inscribed within a circle of radius **R** with equal side length **s**
  * each side also spans an arc of angle Θ
  * we can easily calculate the side **s** via [General Pythagorean](math:trigonometry)
    * c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> - 2ab * cos(Θ) where c is the side opposite Θ
  * since the triangle is inscribed within a circle of radius **R** both **a** and **b** = **R**
    * c = √(R<sup>2</sup> + R<sup>2</sup> - 2RR * cos(Θ))
      * = √(2R<sup>2</sup> - 2R<sup>2</sup> * cos(Θ))
      * = R√(2 - 2 * cos(Θ))

### Pentagon
  * Θ = 360/5 = 72 degrees (2π/5 radians) 
  * s = R * √(2 - 2 * cos(72-deg)); cos(72) = (-1 + √(5)) / 4 (see [this](https://www.quora.com/How-can-I-get-the-fraction-value-of-cos-72))
    * = R * √(2 - 2 * (-1 + √(5)) / 4) = R * √(2 - (-1 + √(5)) / 2) = R * √( (4 + 1 - √5)/2)
    * = R * √( (5 - √5)/2)
    * ≅ R * 1.17557

### Hexagon
  * Θ = 360/6 = 60 degrees
  * s = R * √(2 - 2 * cos(60-deg)) = R * √(2 - 2 * 0.5) = R * √(1)
    * = R