Polygon formula chart
The names, angle sums, and interior angles of polygons from three to ten sides, plus the formulas that generate them. A polygon interior angles depend only on its number of sides.
Polygons by number of sides
| Name | Sides | Interior angle sum | Each angle (regular) |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Quadrilateral | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
| Heptagon | 7 | 900° | 128.57° |
| Octagon | 8 | 1,080° | 135° |
| Nonagon | 9 | 1,260° | 140° |
| Decagon | 10 | 1,440° | 144° |
Polygon formulas (n = number of sides)
| Quantity | Formula |
|---|---|
| Interior angle sum | (n − 2) × 180° |
| Each interior angle (regular) | (n − 2) × 180° / n |
| Each exterior angle (regular) | 360° / n |
| Number of diagonals | n(n − 3) / 2 |
The interior angles of a polygon add up to (n minus 2) times 180 degrees, because any polygon can be split into n minus 2 triangles. For a regular polygon, divide that sum by n to get each angle. The exterior angles of any polygon always total 360 degrees.
Need shape formulas?
See the Area Formula Chart and the Triangle Formula Chart.
Why the angle sum formula works
Pick any polygon and draw diagonals from one corner. A polygon with n sides splits into exactly n minus 2 triangles, and each triangle contributes 180 degrees. So the interior angles total (n minus 2) times 180. A pentagon gives 3 triangles and 540 degrees; an octagon gives 6 triangles and 1,080 degrees.
Regular polygons
In a regular polygon every side and angle is equal, so each interior angle is the total divided by n. The exterior angles, one at each corner, always sum to 360 degrees no matter how many sides, which means each exterior angle of a regular polygon is simply 360 divided by n.
FAQ
What is the sum of interior angles of a polygon?
(n − 2) × 180 degrees, where n is the number of sides. A hexagon (n=6) sums to 720 degrees.
What is each angle of a regular hexagon?
120 degrees. The total 720 degrees divided by 6 sides gives 120 each.
How many diagonals does a polygon have?
n(n − 3) / 2. A pentagon has 5(2)/2 = 5 diagonals.