About the Polygon Interior Angle Calculator
The interior angles of any simple polygon with n sides add up to (n − 2) × 180°. This polygon interior angle calculator gives that sum for any number of sides, along with each interior angle, each exterior angle and the central angle of a regular polygon, plus the number of diagonals.
It also works backwards. Enter the size of one interior angle, one exterior angle or the total angle sum, and it tells you how many sides the polygon has — the classic textbook question “each interior angle of a regular polygon is 156°, how many sides does it have?”. It is aimed at middle- and high-school geometry, tiling and woodworking projects (miter angles for picture frames and gazebos), and anyone designing regular shapes.
The angle sum holds for every simple polygon, regular or not. The “each angle” figures assume a regular polygon, where all sides and angles are equal; the exterior angles of any convex polygon always total 360°.
With the default inputs, the sum of interior angles is 720 °. Change any value above to recalculate instantly.
How to use the polygon interior angle calculator
- 1Choose whether you know the number of sides, one angle, or the angle sum.
- 2Enter the value.
- 3Read the interior-angle sum, each interior and exterior angle, and the number of diagonals.
- 4If you entered an angle, check the number of sides — a fractional answer means no regular polygon fits.
Formula and method
Pick one vertex of an n-sided polygon and draw diagonals to every other non-adjacent vertex: this splits the shape into n − 2 triangles, each contributing 180°, so the interior angles total (n − 2) × 180°. In a regular polygon every angle is equal, so one interior angle is that sum divided by n.
Each exterior angle is the supplement of the interior angle (180° − interior). Walking once around any convex polygon turns you through a full 360°, so a regular polygon’s exterior angles are 360° ÷ n each; that also equals its central angle. To find n from an angle the calculator inverts these: n = 360 ÷ (180 − interior), n = 360 ÷ exterior, or n = S ÷ 180 + 2.
- n
- Number of sides (and vertices)
- S
- Sum of the interior angles
Worked examples
Hexagon (6 sides)
A hexagon splits into 4 triangles, so its angles total 4 × 180° = 720°. In a regular hexagon each angle is 720 ÷ 6 = 120°, each exterior angle is 60°, and there are 6 × 3 ÷ 2 = 9 diagonals.
Regular octagon (stop sign)
An octagon’s interior angles total 6 × 180° = 1,080°, so each corner of a stop sign is 135°, each exterior angle is 45°, and it has 20 diagonals.
Each interior angle is 156° — how many sides?
The exterior angle is 180° − 156° = 24°, and 360 ÷ 24 = 15, so the polygon is a 15-sided pentadecagon whose angles total 13 × 180° = 2,340°.
Angles add up to 1,440°
1,440 ÷ 180 = 8 triangles, so n = 8 + 2 = 10: a decagon. If regular, each angle is 144° and it has 10 × 7 ÷ 2 = 35 diagonals.
Frequently asked questions
What is the formula for the sum of interior angles of a polygon?+
Sum = (n − 2) × 180°, where n is the number of sides. A pentagon (n = 5) has interior angles totaling 3 × 180° = 540°.
How do you find each interior angle of a regular polygon?+
Divide the angle sum by the number of sides: (n − 2) × 180° ÷ n. A regular pentagon has 540° ÷ 5 = 108° at each corner.
What do the exterior angles of a polygon add up to?+
Always 360° for any convex polygon, whatever the number of sides. In a regular polygon each exterior angle is therefore 360° ÷ n.
How do I find the number of sides from an interior angle?+
Work out the exterior angle (180° minus the interior angle) and divide 360 by it. An interior angle of 140° gives an exterior angle of 40°, so n = 360 ÷ 40 = 9 sides.
Can a regular polygon have an interior angle of 100°?+
No. That would need 360 ÷ 80 = 4.5 sides, and the number of sides must be a whole number. Valid interior angles include 60°, 90°, 108°, 120°, 128.57°, 135° and so on.