Library / Foundations: Maths / FMA-55
Foundations: Maths
Interior angle of a regular polygon
Each interior angle of a regular polygon, whose n sides and angles are all equal.
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- FMA-55
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- v1
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- Foundations: Maths
- Result
- θ [°]
A regular polygon has n equal sides and n equal interior angles, so each interior angle is the angle sum (n − 2) × 180° divided by n. The angle approaches 180° as the number of sides increases, which is why a polygon with many sides looks like a circle. Here n ≥ 3.
Each interior angle of a regular polygon, whose n sides and angles are all equal. It tends to 180° as the number of sides grows.
Calculator
- Sum of all the interior angles, (n − 2) × 180°720 °
- Exterior angle, 360°/n60 °
- Check: interior + exterior (always 180°)180 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A regular 6-sided polygon has interior angles of 120° and exterior angles of 60°, which add to 180°. A regular hexagon has 120° and 60°.
- Tiling a flat surface with regular polygons needs the angles at a point to total 360°: three regular hexagons (3 × 120°) do, which is why hexagons tile.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Number of sides | 3 to 20 | 60 | 162 | 102 | +1 | +8.57 ° |
Sweep of Number of sides
| Number of sides | Interior angle (°) |
|---|---|
| 3 | 60 |
| 5.125 | 109.8 |
| 7.25 | 130.3 |
| 9.375 | 141.6 |
| 11.5 | 148.7 |
| 13.63 | 153.6 |
| 15.75 | 157.1 |
| 17.88 | 159.9 |
| 20 | 162 |
Sweep one variable
Variables and how each one contributes
Interior angle
What the equation gives you. Everything else feeds it.
Number of sides
The number of equal sides, a whole number of at least 3. The slider runs from 3 to 20. Raises each interior angle, with diminishing returns: the angle tends towards 180°. Raising it by 1 (to 7) moves the result by +7.14 %, from 120 to 128.6 °.
Worked example with the values above
- Angle sum: (n − 2) × 180° = (6 − 2) × 180° = 720°.
- Divide by the number of angles: 720° / 6.
- θ = 120°.
Assumptions and limits
- The polygon is regular: all sides and all angles equal.
- Flat (plane) geometry.
Common mistakes
- Using the formula for a polygon whose angles are not equal.
- Forgetting the division by n: (n − 2) × 180° is the sum, not one angle.
Statistics and uncertainty
The formula is exact geometry. When laying out a polygon in practice, an angle error accumulates across the n corners, so the last side will not meet the first unless the errors are small or are spread out.
Where it is used
- Cutting mitres, laying out bolt circles and designing regular faceted shapes.
- Checking whether a regular polygon can tile a plane.
Maths you need
- Polygon angle sum: (n − 2) × 180°
- Regular polygon: all sides and all angles equal