Polygon Exterior Angle Calculator
The exterior angle of a polygon is the turn you make at each corner as you trace its boundary. For a regular polygon, where all corners are identical, every exterior angle equals 360 degrees divided by the number of sides, because one full lap around the shape adds up to a single 360-degree turn. This tool returns each exterior angle, the matching interior angle, and confirms the constant 360-degree total. Enter the number of sides as a whole number and the results update at once. The relationship holds for every regular polygon from the triangle upward.
Exterior angle formula
Each exterior angle = 360 / n degrees
Each interior angle = 180 - 360 / n degrees
Exterior angle sum = 360 degrees (any simple polygon)
Here n is the number of sides. The exterior angle is the turn at each vertex; for a regular polygon all turns are equal, so each is one nth of a full circle.
How exterior angles work
- Tracing any simple polygon once completes a single full turn, so all exterior angles total 360 degrees.
- For a regular polygon, each exterior angle is 360 divided by the side count.
- Interior and exterior angles at a vertex are supplementary, adding to 180 degrees.
- More sides mean smaller exterior angles, approaching zero as the shape nears a circle.
- The side count must be a whole number of three or more, so smaller values are invalid.
Exterior angle: frequently asked questions
What is the formula for the exterior angle of a regular polygon?
Each exterior angle of a regular polygon with n sides is 360 / n degrees. A regular hexagon (n = 6) has exterior angles of 360 / 6 = 60 degrees. The exterior angle is the turn you make at each corner when walking around the shape, and these turns always add up to one full circle of 360 degrees.
Why do exterior angles always sum to 360 degrees?
Walking once around the boundary of any simple polygon and back to your start, you complete exactly one full turn, which is 360 degrees. The exterior angle at each vertex is the amount you turn there, so all the turns must add to 360 degrees regardless of how many sides the polygon has.
How do the interior and exterior angles relate?
At each vertex the interior angle and the exterior angle are supplementary, meaning they add up to 180 degrees. So for a regular polygon the interior angle is 180 minus the exterior angle, equal to 180 - 360 / n degrees. This calculator reports both for convenience.
Does this work for irregular polygons?
The total of all exterior angles is always 360 degrees for any simple polygon, regular or irregular. However, the per-vertex value of 360 / n applies only when all angles are equal, that is, for a regular polygon. In an irregular polygon the individual exterior angles vary while still summing to 360 degrees.
What is the minimum number of sides?
A polygon needs at least three sides, so the smallest case is a triangle with exterior angles of 360 / 3 = 120 degrees each in the equilateral case. This calculator treats any side count below three as invalid, and the side count must be a whole number.
Official sources
- National Institute of Standards and Technology: Digital Library of Mathematical Functions: trigonometric functions.
- National Aeronautics and Space Administration: Geometry reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.