Polygon Angle Calculator

Calculate interior angles, exterior angles, angle sums, and diagonals of any regular polygon.

Enter Number of Sides

Result

Interior Angle
120
degrees
Interior Angle 120°
Exterior Angle 60°
Sum of Interior Angles 720°
Sum of Exterior Angles 360°
Number of Diagonals 9
Number of Triangles 4

Step-by-Step Solution

Interior Angle = (n-2) x 180 / n

Understanding Polygon Angles

A polygon is a closed two-dimensional shape with straight sides. The angles of a polygon are determined by the number of sides. For regular polygons (where all sides and angles are equal), these angles can be calculated precisely using simple formulas.

Key Formulas

Interior Angle

Each interior angle of a regular polygon.

(n - 2) x 180 / n

Exterior Angle

Each exterior angle of a regular polygon.

360 / n

Sum of Interior Angles

Total of all interior angles combined.

(n - 2) x 180

Number of Diagonals

Count of all possible diagonals in the polygon.

n(n - 3) / 2

Common Polygons Reference

Here are the angle measures for some commonly encountered regular polygons:

  • Equilateral Triangle (3 sides): Interior angle = 60 degrees, 0 diagonals
  • Square (4 sides): Interior angle = 90 degrees, 2 diagonals
  • Regular Pentagon (5 sides): Interior angle = 108 degrees, 5 diagonals
  • Regular Hexagon (6 sides): Interior angle = 120 degrees, 9 diagonals
  • Regular Octagon (8 sides): Interior angle = 135 degrees, 20 diagonals
  • Regular Decagon (10 sides): Interior angle = 144 degrees, 35 diagonals

Why Do Interior Angles Increase?

As the number of sides increases, each interior angle gets larger and approaches 180 degrees. This is because the polygon becomes more and more circular. In the limit, a polygon with infinitely many sides would be a circle, and each "angle" would be a straight line (180 degrees).

The Exterior Angle Theorem

A remarkable property of all convex polygons is that the sum of exterior angles is always exactly 360 degrees, regardless of the number of sides. This means each exterior angle of a regular n-gon is simply 360/n degrees. Notice that the interior angle and exterior angle at any vertex always sum to 180 degrees.

Practical Applications

  • Architecture: Designing floor tiles, windows, and building layouts.
  • Engineering: Bolt patterns, gear teeth, and structural designs.
  • Nature: Honeycomb hexagons, crystal structures, and snowflake symmetry.
  • Art: Islamic geometric patterns, tessellations, and mandalas.