the polygon interior angle sum

Walk around the inside corners of any flat polygon and add up how much each one turns. Surprisingly, the total depends only on how many corners there are, not on the polygon's exact shape — a stretched-out pentagon and a squat one both add to the same number. This dependable total is the interior angle sum.

The formula is: for a convex polygon with n sides, the sum of the interior angles equals (n-2)·180 degrees. The cleanest reason is to triangulate. Pick one vertex and draw all the diagonals from it; this slices an n-gon into exactly n-2 triangles, and the diagonals partition every interior angle without leaving any leftover. Since each triangle's three angles sum to 180 degrees (the triangle-angle-sum), the polygon's total is (n-2)·180. Check it: a triangle (n=3) gives (3-2)·180 = 180; a quadrilateral (n=4) gives 360; a pentagon (n=5) gives 540; a hexagon (n=6) gives 720. For a regular n-gon, where all angles are equal, each interior angle is therefore (n-2)·180/n degrees — for example 108 degrees in a regular pentagon.

This is one of the most useful facts in plane geometry: it lets you find a missing angle in a many-sided figure, and it underlies which regular polygons can tile a floor edge-to-edge (only those whose interior angle divides 360 exactly — triangles, squares, hexagons). An honest qualification: the simple formula (n-2)·180 is stated for simple (non-self-crossing) polygons; for a convex polygon every interior angle is genuinely less than 180, while for a concave one the formula still holds for the interior angles provided the reflex corner is measured as an angle greater than 180 degrees (not as its smaller 'outside' partner). Self-intersecting star polygons follow a different rule entirely.