The sum of the interior angles of any convex polygon with n sides is given by the formula (n − 2) × 180°. The idea behind the formula: every polygon can be split into (n − 2) triangles by drawing diagonals from one vertex, and every triangle contributes 180° to the total. In a regular polygon (where all the sides and all the angles are equal), you divide the total by the number of angles n to get the size of each interior angle. For example in a pentagon (n = 5): the total is 540°, and each angle is 108°.
The formula for the sum of the angles in a polygon
For any convex polygon (a polygon with no 'dents' pointing inwards) with n sides, the sum of all the interior angles equals (n − 2) × 180°. n is the number of sides (and also the number of vertices and angles, because they are always equal in a polygon). This formula works for every convex polygon, whether or not the sides are equal to each other — it depends only on the number of sides.
| Number of sides (n) | Name of the polygon | Angle sum (n − 2) × 180° | One angle in a regular polygon |
|---|---|---|---|
| 3 | Triangle | 180° | 60° |
| 4 | Quadrilateral | 360° | 90° |
| 5 | Pentagon | 540° | 108° |
| 6 | Hexagon | 720° | 120° |
| 8 | Octagon | 1,080° | 135° |
| 10 | Decagon | 1,440° | 144° |
Proof: why does the formula work?
The idea behind the proof is simple and elegant: choose one vertex of the polygon, and draw diagonals from it to every other vertex (except the two neighbours next to it, because they are already joined to it by sides). This splits the whole polygon into triangles, with no overlaps and no gaps.
How many triangles do you get? Always exactly n − 2. In a quadrilateral (n = 4) you get 2 triangles. In a pentagon (n = 5) you get 3 triangles. In a hexagon (n = 6) you get 4 triangles. The pattern is always: the number of sides minus 2.
Since every triangle contributes exactly 180° to the angle sum (a basic fact of geometry, which can itself be proved using parallel lines), and all the interior angles of the polygon are shared out exactly among the triangles with no overlap — the sum of all the polygon's angles equals the number of triangles times 180°, that is (n − 2) × 180°.
Worked example: the angle sum of a pentagon (n = 5)
Substitute n = 5 into the formula: sum = (5 − 2) × 180° = 3 × 180° = 540°. If the pentagon is regular (all angles equal), each angle equals 540° ÷ 5 = 108°.
Worked example: the angle sum of a hexagon (n = 6)
Substitute n = 6 into the formula: sum = (6 − 2) × 180° = 4 × 180° = 720°. In a regular hexagon, each angle equals 720° ÷ 6 = 120° — exactly the angle you see in a honeycomb or in hexagonal floor tiles.
The angles of a regular polygon — a separate formula
A regular polygon is a polygon in which all the sides are the same length and all the angles are the same size — for example an equilateral triangle, a square, a regular pentagon and so on. To find the size of one interior angle of a regular polygon, simply divide the sum of all the angles (worked out with the usual formula) by the number of angles n:
One interior angle = (n − 2) × 180° ÷ n
For example, in a square (n = 4, always regular): (4 − 2) × 180° ÷ 4 = 360° ÷ 4 = 90° — exactly as you would expect for every angle of a square. In a regular octagon (n = 8): (8 − 2) × 180° ÷ 8 = 1,080° ÷ 8 = 135°.
You can work this out quickly for any number of sides in the interactive polygon angle calculator, which also draws the polygon and the diagonals that split it into triangles.
Common mistakes
- Forgetting the (n − 2) — a common error is to work out n × 180° instead of (n − 2) × 180°, which gives an answer that is too big.
- Confusing the sum of all the angles with a single angle — the angle sum is (n − 2) × 180°, but one angle of a regular polygon is that sum divided by n. They are two completely different numbers.
- Using the formula on a non-convex polygon — the formula (n − 2) × 180° holds for convex polygons. In a polygon with a 'dent' (a reflex interior angle bigger than 180°) the same triangle-splitting argument needs more care.
- Confusing interior with exterior angles — the sum of the exterior angles of any convex polygon is always 360°, regardless of the number of sides. That is a completely different formula from the interior-angle one, and GCSE questions often use it as the quicker route: exterior angle of a regular polygon = 360° ÷ n, interior angle = 180° − exterior angle.
Practice
Once the formula is clear, it is worth practising on several different polygons — triangle, quadrilateral, pentagon, hexagon and octagon — until working out (n − 2) × 180° becomes automatic. The Year 7 geometry page includes more practice on the properties of polygons and angles.
Frequently asked questions
What is the formula for the sum of the angles in a polygon?
The sum of the interior angles of any convex polygon with n sides equals (n − 2) × 180°. For example in a quadrilateral (n = 4): 2 × 180° = 360°. In a pentagon (n = 5): 3 × 180° = 540°.
How do you work out the angles of a regular polygon?
First work out the sum of all the angles with the formula (n − 2) × 180°, then divide by the number of angles n to get the size of one angle. For example in a regular pentagon: 540° ÷ 5 = 108° for each angle. Alternatively, use the exterior angle: 360° ÷ n, then subtract from 180°.
Why is the angle sum of a polygon (n − 2) times 180?
Because every polygon can be split into (n − 2) triangles by diagonals drawn from one vertex, and every triangle contributes exactly 180° to the total. Adding all the triangles together gives the angle sum of the whole polygon.
What is one angle of a regular hexagon?
In a hexagon (n = 6): the angle sum is (6 − 2) × 180° = 720°. In a regular hexagon, each angle equals 720° ÷ 6 = 120°.
Type in the number of sides and instantly see the angle sum and the split into triangles.
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