Polygon Calculator

Category: Geometry
- April 02, 2025
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Calculate various properties of regular polygons including area, perimeter, interior angles, and more. This calculator supports multiple calculation methods and provides visualisations with step-by-step explanations.

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About Regular Polygons in Geometry

A regular polygon is a polygon that is equiangular (all angles are equal) and equilateral (all sides have the same length). Regular polygons are important in geometry, architecture, design, and many other fields.

Key Properties and Formulas
  • Number of Sides (n): The total count of sides and vertices in the polygon.
  • Side Length (s): The length of each side of the polygon.
  • Perimeter (P): P = n × s (The sum of all side lengths)
  • Interior Angle (θint): θint = (n − 2) × 180° / n (The angle inside the polygon at each vertex)
  • Exterior Angle (θext): θext = 360° / n (The angle formed by extending a side)
  • Inradius (r): r = s / (2 × tan(π/n)) (The radius of the inscribed circle)
  • Circumradius (R): R = s / (2 × sin(π/n)) (The radius of the circumscribed circle)
  • Area (A): A = (n × s²) / (4 × tan(π/n))
Common Regular Polygons
  • Triangle (n=3): Also known as an equilateral triangle.
  • Square (n=4): Has four equal sides and angles of 90°.
  • Pentagon (n=5): Has five equal sides and interior angles of 108°.
  • Hexagon (n=6): Has six equal sides and interior angles of 120°.
  • Octagon (n=8): Has eight equal sides and interior angles of 135°.
  • Decagon (n=10): Has ten equal sides and interior angles of 144°.
  • Dodecagon (n=12): Has twelve equal sides and interior angles of 150°.
Applications

Regular polygons have many practical applications:

  • Architecture: Building layouts, decorative elements, floor plans
  • Design: Logos, patterns, product shapes
  • Engineering: Mechanical parts, gear design, structural elements
  • Urban Planning: Traffic roundabouts, public spaces, park layouts
  • Nature: Many natural structures approximate regular polygons (honeycomb cells, crystal structures)
  • Traffic Signs: Stop signs are regular octagons