Interior Angles Calculator
This calculator finds the sum of all interior angles in any polygon, using the formula (n - 2) * 180 degrees, where n is the number of sides. Enter the number of sides, and the calculator instantly displays the total angle sum. For a regular polygon (all sides and angles equal), the calculator also shows the measure of each individual interior angle: [(n - 2) * 180] / n degrees. A reference table shows the angle sums and individual angles for common polygons from triangles through decagons. Interior angle calculations are essential in geometry, construction, surveying, and architectural design. Understanding polygon angles helps verify that angle measurements are correct and enables solving complex geometric problems.
Interior angle formulas
Sum of interior angles = (n - 2) * 180 degrees
Each angle in regular polygon = [(n - 2) * 180] / n degrees
where n = number of sides
Common polygons and their interior angles
| Polygon | Sides | Sum of angles | Each angle (regular) |
|---|---|---|---|
| Triangle | 3 | 180 degrees | 60 degrees |
| Quadrilateral | 4 | 360 degrees | 90 degrees |
| Pentagon | 5 | 540 degrees | 108 degrees |
| Hexagon | 6 | 720 degrees | 120 degrees |
| Heptagon | 7 | 900 degrees | 128.57 degrees |
| Octagon | 8 | 1,080 degrees | 135 degrees |
| Nonagon | 9 | 1,260 degrees | 140 degrees |
| Decagon | 10 | 1,440 degrees | 144 degrees |
Interior angles calculator: frequently asked questions
What is the sum of interior angles?
The sum of all interior angles in any polygon with n sides is (n - 2) * 180 degrees. For example, a triangle (3 sides) has an angle sum of (3 - 2) * 180 = 180 degrees. A square (4 sides) has an angle sum of (4 - 2) * 180 = 360 degrees. A hexagon (6 sides) has an angle sum of (6 - 2) * 180 = 720 degrees.
What is a regular polygon?
A regular polygon is a polygon where all sides have equal length and all interior angles are equal. For a regular polygon with n sides, each interior angle measures [(n - 2) * 180] / n degrees. For example, each angle in a regular hexagon is (6 - 2) * 180 / 6 = 720 / 6 = 120 degrees.
What is the difference between interior and exterior angles?
Interior angles are measured inside the polygon. Exterior angles are measured outside, between one side extended and the adjacent side. The sum of any interior angle and its adjacent exterior angle is always 180 degrees (they are supplementary). The sum of all exterior angles in any polygon is always 360 degrees.
Why does the angle sum formula work?
Any polygon with n sides can be divided into (n - 2) triangles by drawing diagonals from one vertex. Since each triangle has an angle sum of 180 degrees, the total interior angle sum is (n - 2) * 180 degrees. This formula works for any polygon, whether convex or concave.
What is the difference between a convex and concave polygon?
A convex polygon has all interior angles less than 180 degrees, and no inward-pointing vertices. A concave polygon has at least one interior angle greater than 180 degrees, and at least one inward-pointing vertex. The angle sum formula applies to both types.
Official sources
- Geometry fundamentals: National Institute of Standards and Technology.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.