Library / Foundations: Maths / FMA-56
Foundations: Maths
Exterior angle of a regular polygon
Each exterior angle of a regular polygon: the turn made at each corner when walking round it.
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- FMA-56
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- v1
- Field
- Foundations: Maths
- Result
- θext [°]
Walking round a polygon, the direction turns by the exterior angle at each corner. After a complete circuit the total turn is one full revolution, 360°, so for a regular polygon each exterior angle is 360°/n. The interior angle is 180° minus the exterior angle. Here n ≥ 3.
Each exterior angle of a regular polygon: the turn made at each corner when walking round it. The exterior angles of any simple polygon add to 360°.
Calculator
- Interior angle, 180° minus the exterior angle120 °
- Total turn after one full circuit360 °
- Number of sides for a 15° exterior angle24
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 polygon with 6 sides turns by 60° at each corner, so an interior angle is 120°. A square turns 90° and an octagon 45°.
- A vehicle driving a regular polygon turns by this angle at each corner: halving the exterior angle doubles the number of sides.
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 | 18 | 120 | 102 | +1 | −8.57 ° |
Sweep of Number of sides
| Number of sides | Exterior angle (°) |
|---|---|
| 3 | 120 |
| 5.125 | 70.24 |
| 7.25 | 49.66 |
| 9.375 | 38.4 |
| 11.5 | 31.3 |
| 13.63 | 26.42 |
| 15.75 | 22.86 |
| 17.88 | 20.14 |
| 20 | 18 |
Sweep one variable
Variables and how each one contributes
Exterior 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. Lowers each exterior angle in inverse proportion: more sides means a smaller turn at each corner. Raising it by 1 (to 7) moves the result by −14.3 %, from 60 to 51.43 °.
Worked example with the values above
- A full turn is 360°, shared equally between 6 corners.
- Divide: 360° / 6.
- θext = 60°.
Assumptions and limits
- The polygon is regular, so all exterior angles are equal.
- Flat (plane) geometry and a polygon that does not cross itself.
Common mistakes
- Using the interior angle where the turn angle is wanted.
- Applying 360°/n to an irregular polygon.
Statistics and uncertainty
The formula is exact geometry. In practice, steering a path through n corners, small turning errors add up: after n corners an error of ±δ at each leaves the path rotated by about ±δ√n from its intended heading.
Where it is used
- Steering a path round a regular polygon.
- Spacing n equal features, such as bolts, round a circle.
Maths you need
- Full turn: 360 degrees
- Regular polygon: all sides and all angles equal