Polygon Angle Calculator
Enter the number of sides of a regular polygon and this calculator returns the interior angle, exterior angle, sum of all interior angles, central angle, and the number of diagonals. You can also work in reverse: enter a known interior angle or angle sum to find how many sides the polygon has. Results update instantly as you type.
Polygon angle formulas
For a regular polygon with n sides (where n is a whole number of 3 or more), four angle relationships hold. The sum of all interior angles is (n - 2) x 180 degrees, so a triangle sums to 180 degrees, a quadrilateral to 360 degrees, and so on, adding 180 degrees for each extra side. Each individual interior angle of a regular polygon is that sum divided by n: ((n - 2) x 180) / n. Each exterior angle equals 360 / n degrees, and it always combines with the adjacent interior angle to form a straight line of exactly 180 degrees. The central angle, the angle at the centre of a circumscribed circle between two neighbouring vertices, also equals 360 / n degrees. Finally, the number of diagonals (line segments joining non-adjacent vertices) is n(n - 3) / 2, which grows rapidly: a decagon has 35 diagonals, and a 20-gon has 170.
Reverse-solving: finding sides from a known angle
You can work backwards from a known interior angle to the number of sides. If each interior angle is alpha degrees, then n = 360 / (180 - alpha). For example, an interior angle of 108 degrees gives n = 360 / (180 - 108) = 360 / 72 = 5 sides, confirming a regular pentagon. Not every angle value produces a whole-number answer; the calculator flags those cases. You can also start from the angle sum: n = (sum / 180) + 2, so a sum of 1080 degrees gives n = 1080 / 180 + 2 = 6 + 2 = 8 sides, a regular octagon. These reverse modes are useful in geometry proofs, tiling design, and identifying unknown polygons from measured angles.
Interior vs exterior vs central angles
The interior angle is measured inside the polygon between two adjacent sides. The exterior angle is the supplement of the interior angle, measured outside the polygon at each vertex. For any convex polygon, no matter how many sides, all exterior angles sum to exactly 360 degrees - one full turn around the perimeter. The central angle is formed at the geometric centre of a regular polygon by two radii drawn to adjacent vertices; it equals 360 / n degrees and is identical in value to the exterior angle for regular polygons. Understanding the difference matters in construction, navigation, and engineering: a regular hexagonal nut has interior angles of 120 degrees and exterior angles of 60 degrees, which is why a 60-degree wrench fits perfectly.
Diagonals, triangulation, and why the formulas work
Every polygon can be cut into triangles by drawing diagonals from a single vertex, producing exactly (n - 2) triangles. Since each triangle contains 180 degrees, the total interior angle sum is (n - 2) x 180 degrees - this is the geometric proof behind the formula. The full diagonal count n(n - 3) / 2 comes from choosing 2 vertices from n (giving n(n - 1) / 2 combinations), then subtracting the n sides. Diagonals matter in structural engineering: a triangulated frame (where every polygon is subdivided into triangles by diagonals) is rigid, while an un-triangulated quadrilateral frame can deform under load.
Regular polygon angle reference
| Sides | Name | Interior angle | Exterior angle | Angle sum | Diagonals |
|---|---|---|---|---|---|
| 3 | Triangle | 60.00 deg | 120.00 deg | 180 deg | 0 |
| 4 | Quadrilateral | 90.00 deg | 90.00 deg | 360 deg | 2 |
| 5 | Pentagon | 108.00 deg | 72.00 deg | 540 deg | 5 |
| 6 | Hexagon | 120.00 deg | 60.00 deg | 720 deg | 9 |
| 7 | Heptagon | 128.57 deg | 51.43 deg | 900 deg | 14 |
| 8 | Octagon | 135.00 deg | 45.00 deg | 1080 deg | 20 |
| 9 | Nonagon | 140.00 deg | 40.00 deg | 1260 deg | 27 |
| 10 | Decagon | 144.00 deg | 36.00 deg | 1440 deg | 35 |
| 11 | Hendecagon | 147.27 deg | 32.73 deg | 1620 deg | 44 |
| 12 | Dodecagon | 150.00 deg | 30.00 deg | 1800 deg | 54 |
Properties of regular polygons with 3 to 12 sides. Individual angle values assume all sides and angles are equal.
Frequently asked questions
What is the interior angle of a regular polygon?
The interior angle is the angle formed inside the polygon at each vertex, between the two sides meeting there. For a regular polygon with n sides, every interior angle is equal and measures ((n - 2) x 180) / n degrees. A regular triangle has 60 degrees, a square has 90 degrees, a regular pentagon has 108 degrees, and a regular hexagon has 120 degrees.
Why do exterior angles always sum to 360 degrees?
As you walk around the perimeter of any convex polygon and turn at each vertex, you make one full rotation of 360 degrees by the time you return to the start. Each turn equals one exterior angle. So no matter how many sides the polygon has, the exterior angles always add up to 360 degrees. This fact holds for all convex polygons, not just regular ones.
How do I find the number of sides from a known interior angle?
Use the formula n = 360 / (180 - alpha), where alpha is the interior angle in degrees. For example, if each angle is 135 degrees, n = 360 / (180 - 135) = 360 / 45 = 8 sides, which is a regular octagon. The interior angle must be strictly between 60 and 180 degrees for the result to be a valid polygon.
What is the central angle of a polygon?
The central angle is the angle at the exact centre of a regular polygon between two radii drawn to adjacent vertices. It equals 360 / n degrees. For a regular hexagon (n = 6), the central angle is 60 degrees. The central angle equals the exterior angle in any regular polygon, which is useful in both geometry and design.
How many diagonals does a polygon have?
The formula is n(n - 3) / 2, where n is the number of sides. A triangle has 0 diagonals, a quadrilateral has 2, a pentagon has 5, a hexagon has 9, and a decagon has 35. The formula counts the pairs of vertices, subtracts the n edges, and halves the result to avoid double-counting.
Do these formulas apply to irregular polygons?
The sum of interior angles formula, (n - 2) x 180 degrees, applies to any simple (non-self-intersecting) polygon, whether regular or irregular. However, the individual angle formulas assume a regular polygon where all sides and all angles are equal. An irregular pentagon still has a 540-degree angle sum, but each angle can be different.
As the number of sides increases, what happens to the angles?
As n grows, each interior angle approaches 180 degrees and each exterior angle approaches 0 degrees. A 1000-sided polygon (chiliagon) has an interior angle of 179.64 degrees. In the limit, an infinite-sided regular polygon is a circle, where the "interior angle" at any point on the boundary reaches 180 degrees (a straight line) and the "exterior angle" is infinitesimally small.