Polygon Angle Calculator
Enter the number of sides to get the sum of the interior angles, each interior and exterior angle of the regular polygon, its central angle and its diagonals — in degrees and in radians as multiples of π. Or work backward: find the number of sides from an angle, or the missing angle of any polygon. Free, no sign-up; it runs in your browser.
Results
108°
Each interior angle (regular)
72°
Each exterior angle (regular)
5
Diagonals
| Angle | Degrees | Radians |
|---|---|---|
| Sum of the interior angles | 540° | 3π ≈ 9.4248 |
| Each interior angle (regular) | 108° | 3π/5 ≈ 1.885 |
| Each exterior angle (regular) | 72° | 2π/5 ≈ 1.2566 |
| Sum of the exterior angles | 360° | 2π ≈ 6.2832 |
| Central angle (regular) | 72° | 2π/5 ≈ 1.2566 |
The working
Σ interior = (n − 2) × 180° = (5 − 2) × 180° = 540°
Each interior = (n − 2) × 180° / n = 540° / 5 = 108°
Each exterior = 360° / n = 360° / 5 = 72°
Interior + exterior = 108° + 72° = 180°
Diagonals = n(n − 3) / 2 = 5 × 2 / 2 = 5
Figure
α is an interior angle, β an exterior angle (between a side extended and the next side) and γ the central angle.
How to Use the Polygon Angle Calculator
- 1. Say what you know. The number of sides, one interior angle or one exterior angle of a regular polygon, or all but one angle of any polygon.
- 2. Enter the value. Angles can be typed in degrees, or in radians as a multiple of π such as 3pi/5. The number of sides is a whole number from 3 to 1,000.
- 3. Read the table. The angle sums and each angle of the regular polygon are shown in degrees and in radians, as exact fractions where the answer is not a whole number: a heptagon's interior angle is 900/7°, not a rounded 128.57°.
- 4. Check the working and the figure. Every formula is shown with your numbers in it, beside a regular polygon with an interior, an exterior and a central angle marked.
Polygon Angle Formulas
| Quantity | Formula | Applies to |
|---|---|---|
| Sum of the interior angles | (n − 2) × 180° | Any simple polygon with n sides |
| Each interior angle | (n − 2) × 180° / n | Regular polygons |
| Each exterior angle | 360° / n | Regular polygons |
| Sum of the exterior angles | 360° | Any convex polygon, one per vertex |
| Central angle | 360° / n | Regular polygons |
| Number of diagonals | n(n − 3) / 2 | Any convex polygon |
| Sides from an interior angle | n = 360° / (180° − interior) | Regular polygons |
| Sides from an exterior angle | n = 360° / exterior | Regular polygons |
The sum comes from cutting the polygon into triangles by drawing every diagonal from one corner: that makes n − 2 triangles, each with an angle sum of 180°. An interior and an exterior angle at the same corner are supplementary — they make a straight line — so each pair adds up to 180°.
| Polygon | Sides | Sum of interior angles | Each interior angle | Each exterior angle |
|---|---|---|---|---|
| Triangle | 3 | 180° | 60° | 120° |
| Quadrilateral | 4 | 360° | 90° | 90° |
| Pentagon | 5 | 540° | 108° | 72° |
| Hexagon | 6 | 720° | 120° | 60° |
| Octagon | 8 | 1080° | 135° | 45° |
| Decagon | 10 | 1440° | 144° | 36° |
| Dodecagon | 12 | 1800° | 150° | 30° |
Worked Examples
- An octagon. n = 8: the interior angles sum to (8 − 2) × 180° = 1080°, each is 1080° / 8 = 135°, each exterior angle is 360° / 8 = 45°, and it has 8 × 5 / 2 = 20 diagonals.
- Sides from an interior angle. A regular polygon with 140° angles has n = 360° / (180° − 140°) = 9 sides: a nonagon.
- No such polygon. An interior angle of 100° would need n = 360° / 80° = 4.5 sides, so no regular polygon has it.
- A missing angle. A quadrilateral with angles 90°, 80° and 100° has 360° − 270° = 90° left for the fourth.
- In radians. A regular pentagon's interior angle is 108° = 3π/5 rad, and its exterior angle is 72° = 2π/5 rad.
Regular and Irregular Polygons
The sum formulas hold for any simple polygon — one whose sides do not cross — whether or not it is regular, and even when it is concave. The "each angle" formulas need all the angles to be equal, so they only describe regular polygons. In a concave polygon one interior angle exceeds 180° (a reflex angle), which the missing-angle mode will flag.
Frequently Asked Questions
- What is the formula for the sum of the interior angles of a polygon?
- (n − 2) × 180°, where n is the number of sides. A triangle gives 180°, a quadrilateral 360° and a pentagon 540°. The formula comes from splitting the polygon into n − 2 triangles.
- How do you find each interior angle of a regular polygon?
- Divide the sum by the number of sides: (n − 2) × 180° / n. For a regular hexagon that is 720° / 6 = 120°. Equivalently, each interior angle is 180° minus the exterior angle 360° / n.
- What is the sum of the exterior angles of a polygon?
- 360°, for any convex polygon, whatever the number of sides — taking one exterior angle at each vertex. Walking around the polygon, you turn through a total of exactly one full turn.
- How do you find the number of sides from an interior angle?
- Use n = 360° / (180° − interior angle), which is 360° divided by the exterior angle. If the result is a whole number of at least 3, that regular polygon exists: 150° gives n = 12, a dodecagon.
- Why is there no regular polygon with an interior angle of 100°?
- Because the exterior angle would be 80°, and 360° / 80° = 4.5. A polygon has a whole number of sides, so the angle that would close a regular polygon after 4.5 turns of 80° does not exist.
- How many diagonals does a polygon have?
- n(n − 3) / 2. Each of the n corners connects to n − 3 others by a diagonal (not itself and not its two neighbors), and dividing by 2 stops each diagonal being counted from both ends. A hexagon has 9, a decagon 35.
- Do these formulas work in radians?
- Yes, replace 180° with π. The interior angles of an n-gon sum to (n − 2)π, each interior angle of a regular polygon is (n − 2)π / n, and each exterior angle is 2π / n. The calculator shows both.