Polygon angle calculator
Interior, exterior and missing angles
Add a side length to get the size too. Any unit.
Result
Interior angle
120.00°
- Exterior angle
- 60.00°
- Central angle
- 60.00°
- Sum of the interior angles
- 720°
- Diagonals
- 9
About this tool
Find the interior, exterior and central angles of any regular polygon, work out how many sides an angle implies, or find the missing angle of any polygon — with area and radii from a side length.
The polygon formulas
Any polygon with n sides can be split into n − 2 triangles from one corner, so its interior angles always add up to (n − 2) × 180°. In a regular polygon, where every side and angle is equal, the rest follows:
- Each interior angle = (n − 2) × 180° ÷ n.
- Each exterior angle = 360° ÷ n, the turn at every corner as you walk round the edge. The central angle, between two corners seen from the middle, is the same.
- Number of diagonals = n × (n − 3) ÷ 2.
- From a side length s: apothem = s ÷ (2 tan(180° ÷ n)), circumradius = s ÷ (2 sin(180° ÷ n)), and area = perimeter × apothem ÷ 2.
Working backwards from an angle
Since each exterior angle is 360° ÷ n, the number of sides is 360° ÷ (180° − interior angle). An interior angle of 150° means a regular 12-sided polygon, and an exterior angle of 24° means 15 sides. If the answer is not a whole number, no regular polygon has that angle, and the calculator shows the closest ones that exist.
A missing angle
For any polygon, regular or not, subtract the angles you know from (n − 2) × 180°. This only works when the polygon is simple, with no edges crossing, and every angle is less than 360°.
To build a frame or box with this many sides, the miter calculator gives the saw settings: 180° ÷ n on each end of every piece.
Frequently asked questions
What are the angles of a pentagon?
Each interior angle of a regular pentagon is 108°, and the five interior angles add up to 540°. Each exterior angle is 72°.
The interior angles of any pentagon, regular or not, add up to 540°, and the calculator above finds a missing one. To build a five-sided frame, cut both ends of every piece at 36°: see the miter settings for five sides.
What are the angles of a hexagon?
Each interior angle of a regular hexagon is 120°, and the six interior angles add up to 720°. Each exterior angle is 60°.
These come from a formula that works for any polygon with n sides: the interior angles add up to (n − 2) × 180°, so in a regular polygon each one is (n − 2) × 180° ÷ n. A hexagonal frame is cut at 30° on each end: see the miter settings for six sides.
What are the angles of an octagon?
Each interior angle of a regular octagon, such as a stop sign, is 135°, and the eight interior angles add up to 1,080°. Each exterior angle is 45°.
An eight-sided frame or box is cut at 22.5° on each end: see the miter settings for eight sides. For any other number of sides, change the number above.
Guides
About these measurements
Readings are estimates taken from rendered pixels and consumer device sensors. Do not rely on them for structural engineering, surveying, manufacturing inspection, navigation, or medical decisions. Check anything safety-critical with a calibrated instrument.
All processing happens in your browser. No image, camera frame or sensor reading is ever uploaded.
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