https://github.com/jaytwolab/polygon.circle

Regular n-pronged circumcircle and an inscribed circle :kr: 정n각형의 외접원과 내접원
https://github.com/jaytwolab/polygon.circle

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Regular n-pronged circumcircle and an inscribed circle :kr: 정n각형의 외접원과 내접원

• Host: GitHub
• URL: https://github.com/jaytwolab/polygon.circle
• Owner: JayTwoLab
• License: mit
• Created: 2024-12-20T01:38:20.000Z (5 months ago)
• Default Branch: main
• Last Pushed: 2025-02-19T11:42:48.000Z (3 months ago)
• Last Synced: 2025-03-28T03:07:33.747Z (about 2 months ago)
• Language: Python
• Homepage:
• Size: 46.9 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

        
## Regular n-gon Geometry: Circumscribed and Inscribed Circles

> [English](README.md), [Korean](README.ko.md)

  


- A **regular n-gon** is a polygon with all sides and angles equal. It can have both a circumscribed circle (circumcircle) and an inscribed circle (incircle).

   - The **circumcircle** passes through all vertices of the n-gon.

   - The **incircle** is tangent to all sides of the n-gon.

### Circumradius (R) of a Regular n-gon

- For a regular n-gon with side length [![equation](https://latex.codecogs.com/png.latex?%5C%28%20a%20%5C%29)](#) , the radius [![equation](https://latex.codecogs.com/png.latex?%5C%28%20R%20%5C%29)](#) of the circumcircle is calculated as:

   \[

   [![equation](https://latex.codecogs.com/png.latex?R%20%3D%20%5Cfrac%7Ba%7D%7B2%20%5Ccdot%20%5Csin%5Cleft%28%5Cfrac%7B%5Cpi%7D%7Bn%7D%5Cright%29%7D)](#)

   \]

### Inradius (r) of a Regular n-gon

- The radius [![equation](https://latex.codecogs.com/png.latex?%5C%28%20r%20%5C%29)](#) of the incircle is calculated as:

   \[

   [![equation](https://latex.codecogs.com/png.latex?r%20%3D%20%5Cfrac%7Ba%7D%7B2%20%5Ccdot%20%5Ctan%5Cleft%28%5Cfrac%7B%5Cpi%7D%7Bn%7D%5Cright%29%7D)](#)

   \]

### Example: Regular Pentagon (n = 5)

- For a regular pentagon with side length [![equation](https://latex.codecogs.com/png.latex?%5C%28%20a%20%5C%29)](#) :

   - **Circumradius**: [![equation](https://latex.codecogs.com/png.latex?%5C%28%20R%20%3D%20%5Cfrac%7Ba%7D%7B2%20%5Ccdot%20%5Csin%5Cleft%28%5Cfrac%7B%5Cpi%7D%7B5%7D%5Cright%29%7D%20%5C%29)](#)

   - **Inradius**: [![equation](https://latex.codecogs.com/png.latex?%5C%28%20r%20%3D%20%5Cfrac%7Ba%7D%7B2%20%5Ccdot%20%5Ctan%5Cleft%28%5Cfrac%7B%5Cpi%7D%7B5%7D%5Cright%29%7D%20%5C%29)](#)

- These formulas are useful for understanding the geometric properties of regular polygons and have applications in various fields.