Polygon Interior Angle Calculator
Calculate the interior and exterior angles of a regular polygon.
The angles of a many-sided shape
A regular polygon has equal sides and equal angles, and its angles depend only on the number of sides. This calculates the interior and exterior angles from the number of sides, for any regular polygon from a triangle upward.
Enter the number of sides to find the angles.
The elegant rule behind polygon angles
There is a beautifully simple fact at the heart of this: the exterior angles of any polygon always add up to a full turn, 360 degrees, no matter how many sides it has. Divide that by the number of sides and you have each exterior angle of a regular polygon, and the interior angle is what remains on a straight line. This is why the interior angles grow as a polygon gains sides, approaching a straight line as it starts to resemble a circle. The rule explains which regular polygons tile a flat surface without gaps — only those whose angles divide evenly into a full turn, which is why floors are tiled with triangles, squares and hexagons but not pentagons. Simple angle facts turn out to govern the geometry of tiling and structure.
Common questions
What do the exterior angles of a polygon add up to?
Always 360 degrees, a full turn, no matter how many sides — dividing by the number of sides gives each exterior angle of a regular polygon.
Why do only some polygons tile a floor?
Only those whose angles divide evenly into a full turn tile without gaps, which is why triangles, squares and hexagons work but pentagons do not.
Is anything uploaded?
No. The calculation happens in your browser.