math:geometry
Table of Contents
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 * √( (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.txt · Last modified: 2025/12/01 22:05 by ron