Regular n-gon Geometry: Circumscribed and Inscribed Circles

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• 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 , the radius of the circumcircle is calculated as: [ ]

Inradius (r) of a Regular n-gon

• The radius of the incircle is calculated as: [ ]

Example: Regular Pentagon (n = 5)

• For a regular pentagon with side length :

  • Circumradius:
  • Inradius:

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