What This Calculator Does
This tool calculates the sum of the interior angles of any simple polygon based on its number of sides. A polygon's interior angles always add up to a fixed total that depends only on how many sides it has — not on its shape or size. Enter the number of sides and you instantly get the total, plus the measure of each angle if the polygon is regular (all sides and angles equal).
The Formula
The sum of interior angles is given by:
$$S = (n - 2) \times 180^\circ$$
Here n is the number of sides. The logic: any convex polygon can be split into \((n - 2)\) triangles, and each triangle's angles sum to \(180^\circ\). For a regular polygon, each interior angle equals \(S \div n\). The exterior angles of any convex polygon always sum to \(360^\circ\), so each exterior angle of a regular polygon is \(360^\circ \div n\).
How to Use It
Type the number of sides (must be 3 or more) and read the results. For example, a pentagon has 5 sides.
Worked Example
For a hexagon, \(n = 6\). Sum of interior angles = $$(6 - 2) \times 180 = 4 \times 180 = 720^\circ$$ If it is regular, each interior angle = \(720 \div 6 = 120^\circ\), and each exterior angle = \(360 \div 6 = 60^\circ\).
FAQ
Does this work for irregular polygons? Yes — the sum depends only on the number of sides. The "each angle" values, however, assume the polygon is regular.
What about a triangle? \(n = 3\) gives \((3 - 2) \times 180 = 180^\circ\), the familiar triangle angle sum.
Why do exterior angles always add to 360°? Walking once around any convex polygon turns you through a full circle, totalling \(360^\circ\) of turns regardless of the number of sides.
