Polygon Interior Angle Sum Calculator
Every simple polygon, from a triangle to a hundred-sided shape, has interior angles that add up to a fixed total set only by the number of sides. The rule comes from splitting the polygon into triangles, each contributing 180 degrees. This tool computes the total interior angle sum for any polygon and, for the regular case where all angles are equal, the size of each individual angle. Enter how many sides the polygon has as a whole number. The exterior angle sum, always 360 degrees for any simple polygon, is shown alongside for completeness.
Interior angle sum formula
Interior angle sum = (n - 2) * 180 degrees
Each interior angle (regular) = (n - 2) * 180 / n degrees
Exterior angle sum = 360 degrees (any simple polygon)
Here n is the number of sides. Splitting a polygon into n - 2 triangles, each holding 180 degrees, gives the total. The exterior angles of any simple polygon always sum to a single full turn.
How the interior angle sum works
- A polygon with n sides divides into exactly n - 2 triangles from one vertex.
- Each triangle holds 180 degrees, so the total is (n - 2) times 180 degrees.
- The total is identical for regular and irregular polygons with the same side count.
- Dividing by n gives the equal angle of a regular polygon only.
- The side count must be a whole number of three or more, so smaller values are invalid.
Interior angle sum: frequently asked questions
What is the formula for the sum of interior angles of a polygon?
The sum of the interior angles of a polygon with n sides is (n - 2) * 180 degrees. For a triangle (n = 3) the sum is 180 degrees; for a quadrilateral (n = 4) it is 360 degrees; for a pentagon (n = 5) it is 540 degrees. The rule holds for any simple polygon, regular or irregular.
Why is the formula (n minus 2) times 180?
Any simple polygon with n sides can be split into n - 2 triangles by drawing diagonals from a single vertex. Each triangle contributes 180 degrees of interior angle, so the total is (n - 2) * 180. This triangulation argument works for both convex and concave simple polygons.
Does this also give each interior angle of a regular polygon?
Yes. For a regular polygon, where all angles are equal, divide the total by the number of sides: each interior angle equals (n - 2) * 180 / n degrees. This calculator reports both the total sum and the single regular-polygon angle for convenience.
Does the formula work for irregular polygons?
The total sum (n - 2) * 180 is the same for any simple polygon with n sides, whether regular or irregular, because triangulation does not depend on the side lengths. Only the per-angle value differs: in an irregular polygon the individual angles vary while still adding up to the same total.
What is the smallest polygon this applies to?
The smallest polygon is the triangle with three sides, giving a sum of 180 degrees. A shape needs at least three sides to enclose an area, so this calculator treats any side count below three as invalid. The side count must also 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.