How to Use the Polygon Interior & Exterior Angle Sum Calculator
Just enter the number of vertices (n) and instantly calculate that polygon's interior and exterior angle sums. The interior angle sum follows the formula 180 x (n - 2): a triangle (n=3) is 180 degrees, a quadrilateral (n=4) is 360 degrees, and a pentagon (n=5) is 540 degrees, increasing by 180 degrees for each added vertex.
This formula works because drawing diagonals from a single vertex splits an n-sided polygon into (n - 2) triangles. Since every triangle's interior angles always sum to 180 degrees, the total interior angle sum equals 180 degrees times the number of triangles.
The exterior angle sum, on the other hand, is always exactly 360 degrees regardless of how many sides the polygon has, because walking once around the perimeter turns your direction through exactly one full rotation. For a regular polygon, dividing each sum by n also gives you the size of a single interior or exterior angle. Handy for checking geometry homework or working out angles in architecture and design.
Frequently Asked Questions
Walking once around the perimeter of a convex polygon turns your direction through exactly one full rotation (360 degrees), and the turn at each vertex is its exterior angle. That total turning is always 360 degrees no matter how many vertices the polygon has.
Drawing diagonals from one vertex of an n-sided polygon splits it into (n-2) triangles. Since each triangle's interior angles sum to 180 degrees, the polygon's total interior angle sum is 180 times (n-2).