Sum of Interior Angles Calculator

How it is calculated

Enter the number of sides n, which must be a whole number of at least 3, since a polygon needs three sides. The tool uses sum = (n − 2) × 180 degrees: from one corner you can draw n − 3 diagonals, which split the polygon into n − 2 triangles, and each triangle contributes 180°. It also shows each angle = sum ÷ n, which is the size of one interior angle only if the polygon is regular (all sides and angles equal). For an irregular polygon the angles add up to the same total but differ individually. A fractional number of sides such as 5.5, or fewer than 3 sides, shows no result rather than being rounded. The rule is for simple polygons, convex or concave, whose sides do not cross.

Common mistakes

• Using n × 180 or (n − 1) × 180 instead of (n − 2) × 180. A triangle with n = 3 must give 180, not 540.
• Reading the "each angle" value for an irregular polygon. It is just the average, since only a regular polygon has all angles equal.
• Confusing interior angles with exterior angles. The exterior angles of any convex polygon add up to 360° regardless of the number of sides, while the interior sum grows with n.
• Entering a fractional number of sides, such as 5.5, or fewer than 3. Neither is a polygon, so the tool shows no result; enter a whole number of 3 or more.
• Applying the formula to a star-shaped polygon whose edges cross, such as a pentagram; the (n − 2) × 180 rule is for simple polygons.