math:geometry

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Table of Contents

Polygons
- Regular Polygons
  - Pentagon

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
- c² = a² + b² - 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 = √(R² + R² - 2RR * cos(Θ))
  - = √(2R² - 2R² * 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
= R * √(2 - 2 * (-1 + √(5)) / 4) = R * √(2 - (-1 + √(5)) / 2) = R * √¹⁾

¹⁾

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

math/geometry.1764626458.txt.gz · Last modified: 2025/12/01 22:00 by ron