The Polygon Angle Calculator calculates the angles of polygons from the number of sides or from the angles you already know.
For a regular polygon, each interior angle is (n − 2) × 180° ÷ n and each exterior angle is 360° ÷ n. Other modes find the missing angle of a triangle or four-sided shape, and all three angles of a triangle from its side lengths using the law of cosines.
Polygon Angle Calculator
Interior and exterior angles of regular polygons, the angle sum of any polygon, the missing angle of a triangle or 4-sided shape, and triangle angles from three sides.
Formula
- Sum of interior angles: S = (n − 2) × 180°
- Each interior angle (regular): (n − 2) × 180° ÷ n
- Each exterior angle (regular): 360° ÷ n
- Number of diagonals: n(n − 3) ÷ 2
- Missing triangle angle: 180° − (A + B)
- Missing quadrilateral angle: 360° − (A + B + C)
- Triangle angle from sides: cos A = (b² + c² − a²) ÷ (2bc)
Where n = number of sides; A, B, C = angles; a, b, c = sides opposite those angles.
How the Calculation Works
Step 1: Any polygon can be split into (n − 2) triangles from one corner, and each triangle holds 180°. That gives the angle sum.
Step 2: In a regular polygon all angles are equal, so divide the sum by n.
Step 3: Each exterior angle is 180° minus the interior angle; the exterior angles of any convex polygon always add up to 360°.
Variables Explained
| Term | Meaning |
|---|---|
| Interior angle | Angle inside the polygon at a corner |
| Exterior angle | Angle between one side and the extension of the next side |
| Diagonal | Line joining two corners that are not next to each other |
Example Calculation
Regular hexagon
Given: n = 6
Result: sum = 720°; each interior angle = 120°; each exterior angle = 60°; diagonals = 9
Missing angle of a quadrilateral
Given: 85°, 95° and 100°
Result: 360° − 280° = 80°
Triangle angles from sides 5, 6 and 7
Result: A ≈ 44.42°, B ≈ 57.12°, C ≈ 78.46°
Polygon Angle Table
| Polygon | Sides | Angle sum | Interior angle (regular) | Exterior angle |
|---|---|---|---|---|
| Triangle | 3 | 180° | 60° | 120° |
| Quadrilateral | 4 | 360° | 90° | 90° |
| Pentagon | 5 | 540° | 108° | 72° |
| Hexagon | 6 | 720° | 120° | 60° |
| Octagon | 8 | 1,080° | 135° | 45° |
| Decagon | 10 | 1,440° | 144° | 36° |
| Dodecagon | 12 | 1,800° | 150° | 30° |
Shape Properties
The angle-sum rule works for every polygon, regular or not, as long as it is simple (sides don’t cross). The per-angle formulas apply only when all angles are equal.
Dimensions Explained
Angles are entered and shown in degrees. For the side-lengths mode, the sides must be able to form a triangle (each pair longer than the third side).
Frequently Asked Questions
How do you find the interior angle of a polygon?
For a regular polygon: (n − 2) × 180° ÷ n.
What is the sum of the interior angles of a polygon?
(n − 2) × 180°. A hexagon’s angles add up to 720°.
How do you find the exterior angle of a regular polygon?
360° ÷ n.
How do you find the number of sides from an interior angle?
n = 360° ÷ (180° − interior angle). An interior angle of 140° means 9 sides.
How do you find a missing angle in a 4-sided shape?
Subtract the three known angles from 360°.