Every polygon — triangle, quadrilateral, pentagon, hexagon, on up to a 100-gon — has predictable angle formulas based on the number of sides. Two facts to remember and you can solve every polygon angle problem ever set:
| Formula | Equation | Use When |
|---|---|---|
| Sum of interior angles | S = (n − 2) × 180° | Any polygon, regular or not |
| Each interior angle (regular only) | a = (n − 2) × 180° / n | All sides + angles equal |
| Sum of exterior angles | 360° (always) | Any convex polygon |
| Each exterior angle (regular only) | e = 360° / n | All sides equal |
Bonus identity: at any vertex, interior + exterior = 180° (they’re supplementary).
Pick any polygon and draw all diagonals from one vertex. You’ll always carve it into exactly n − 2 triangles. Each triangle’s three angles sum to 180°, and together their angles fill the entire polygon. So:
Polygon angle sum = (n − 2) triangles × 180° per triangle = (n − 2) × 180°
This is the most important geometry derivation to understand — once you see WHY, you’ll never forget the formula.
| n (sides) | Name | Interior sum | Each interior (regular) | Each exterior (regular) |
|---|---|---|---|---|
| 3 | Triangle | 180° | 60° | 120° |
| 4 | Quadrilateral | 360° | 90° | 90° |
| 5 | Pentagon | 540° | 108° | 72° |
| 6 | Hexagon | 720° | 120° | 60° |
| 7 | Heptagon | 900° | ≈ 128.57° | ≈ 51.43° |
| 8 | Octagon | 1080° | 135° | 45° |
| 9 | Nonagon | 1260° | 140° | 40° |
| 10 | Decagon | 1440° | 144° | 36° |
| 11 | Hendecagon | 1620° | ≈ 147.27° | ≈ 32.73° |
| 12 | Dodecagon | 1800° | 150° | 30° |
If you know the interior angle sum S, the number of sides is:
n = S / 180° + 2
Example: S = 1980° → n = 1980/180 + 2 = 11 + 2 = 13 sides (tridecagon).
For a regular polygon: a = (n − 2) × 180° / n. Solve for n:
n = 360° / (180° − a)
Example: each interior angle is 162°. n = 360 / (180 − 162) = 360 / 18 = 20 sides (icosagon).
Pentagon with 4 known angles (110°, 95°, 130°, 105°). Sum = (5 − 2) × 180° = 540°. Missing angle = 540° − (110 + 95 + 130 + 105) = 540° − 440° = 100°.
“Each interior angle of a regular polygon is 144°. How many sides?” Use n = 360/(180 − 144) = 360/36 = 10 sides (decagon).
“In a hexagon, four angles are 120° each. The remaining two are equal. Find them.” Sum = 720°. Known = 4 × 120 = 480°. Remaining 2 sum to 720 − 480 = 240°. Each = 120°.
Imagine walking around the polygon. At each vertex you turn by the exterior angle. After completing the loop, you’ve turned a full 360°. This is true for ANY convex polygon — n could be 3, 100, or 1000, the total turn is always 360°.
This makes the per-vertex exterior angle formula trivially e = 360°/n for regular polygons.
For an interactive tool, use our Polygon Angle Sum Calculator — enter n and get all four values at once. For finding n from a known sum or angle, try our Polygon Sides Calculator.
Do these formulas work for concave polygons? Yes for the interior sum (still (n−2)×180°). For exterior angles, “concave” can have negative or reflex exterior angles which still sum to 360° if you account for sign correctly. Most school problems use convex polygons.
What about star polygons? Star polygons (pentagram, etc.) follow different rules — the formula above is for simple convex/concave polygons only.
Can I use radians? Yes. Replace 180° with π. Sum = (n − 2)π, exterior sum = 2π. Most school work uses degrees.