Regular Polygon Calculator

Calculate area, perimeter, apothem, angles, diagonals, and circumradius of a regular polygon with step-by-step solutions.

Enter Polygon Properties

Results

Area
--
square units
Perimeter --
Apothem --
Circumradius --
Interior Angle --
Exterior Angle --
Sum of Interior Angles --
Number of Diagonals --
Polygon Name --

Step-by-Step Solution

What Is a Regular Polygon?

A regular polygon is a closed plane figure with all sides of equal length and all interior angles of equal measure. Examples include the equilateral triangle (3 sides), square (4 sides), regular pentagon (5 sides), and regular hexagon (6 sides). The more sides a regular polygon has, the more it resembles a circle.

Key Formulas

Area

From side count and side length.

A = (ns²) / (4 tan(π/n))

Perimeter

Sum of all sides.

P = n × s

Apothem

Distance from center to midpoint of a side.

a = s / (2 tan(π/n))

Circumradius

Distance from center to a vertex.

R = s / (2 sin(π/n))

Interior Angle

Each angle inside the polygon.

θ = (n-2) × 180° / n

Diagonals

Number of line segments connecting non-adjacent vertices.

D = n(n-3) / 2

Common Regular Polygons

  • Equilateral Triangle (n=3): Interior angle 60°, simplest regular polygon.
  • Square (n=4): Interior angle 90°, most common in architecture.
  • Pentagon (n=5): Interior angle 108°, found in nature (starfish).
  • Hexagon (n=6): Interior angle 120°, efficient tiling pattern (honeycombs).
  • Octagon (n=8): Interior angle 135°, used for stop signs.
  • Decagon (n=10): Interior angle 144°.
  • Dodecagon (n=12): Interior angle 150°, used in clock faces.

Relationship Between Apothem and Circumradius

The apothem (a) and circumradius (R) are related by: a = R × cos(π/n). The apothem is always shorter than the circumradius. As the number of sides increases, their ratio approaches 1, and both approach the radius of the limiting circle.