# Circle

**Type:** Concept  
**Domain:** Geometry  
**Codex URL:** /mathematics/geometry/circle/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Set of points equidistant from a centre. C=2πr, A=πr².

## Equation
(x−h)²+(y−k)²=r² for circle centred at (h,k), radius r.

## Key theorems
Inscribed angle theorem; Thales' theorem (angle in semicircle = 90°, c.600 BCE); tangent ⊥ radius; power of a point.

## Multi-civilisational π history
Babylon (~1900 BCE, π≈3.125); Egypt (Rhind Papyrus, π≈3.16); Archimedes (c.250 BCE, 223/71<π<22/7); Zu Chongzhi (China, 5th c. CE, 7 decimals); Mādhava (India, c.1400, 11 decimals).

## π is transcendental
Lindemann (1882) — makes squaring the circle by compass/straightedge provably impossible.

## Circle packing
Hexagonal packing density = π/√12 ≈ 90.69% (Thue 1890, rigorous proof Fejes Tóth 1940) — optimal in the plane.

## Non-Euclidean circles
Circumference grows exponentially with radius in hyperbolic geometry; very different behavior in spherical geometry.

## Sources
### Tier 1
- Coxeter, H.S.M. and Greitzer, S.L. (1967). *Geometry Revisited*. MAA.
- Beckmann, P. (1971). *A History of Pi*.
### Tier 2
- Fejes Tóth, L. (1972). *Lagerungen in der Ebene*. 2nd ed.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*