Library / Foundations: Maths / FMA-54
Foundations: Maths
Sum of the interior angles of a polygon
Total of the interior angles of any simple polygon with n sides.
- Ref
- FMA-54
- Rev
- v1
- Field
- Foundations: Maths
- Result
- S [°]
A simple polygon with n ≥ 3 sides can be cut into n − 2 triangles, each with angle sum 180°. The interior angles of any simple (non-crossing) n-sided polygon therefore add up to (n − 2) × 180°. Equal angles require a regular polygon; this entry is the sum for any simple polygon. For n = 3 it gives 180° and for n = 4, 360°.
Total of the interior angles of any simple polygon with n sides. It follows from splitting the shape into n − 2 triangles.
Calculator
- Number of triangles it can be cut into, n − 24
- Interior angle if the polygon is regular120 °
- Exterior angle if the polygon is regular60 °
- Sum of the exterior angles (always 360° for a simple polygon)360 °
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 6-sided polygon has interior angles totalling 720°, from 4 triangles. A hexagon gives 720° and a regular hexagon has six 120° angles.
- Each extra side adds exactly 180° to the total, so the sum grows without limit even though each regular angle approaches 180°.
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 | 180 | 3,240 | 3,060 | +1 | +180 ° |
Sweep of Number of sides
| Number of sides | Sum of interior angles (°) |
|---|---|
| 3 | 180 |
| 5.125 | 562.5 |
| 7.25 | 945 |
| 9.375 | 1,328 |
| 11.5 | 1,710 |
| 13.63 | 2,093 |
| 15.75 | 2,475 |
| 17.88 | 2,858 |
| 20 | 3,240 |
Sweep one variable
Variables and how each one contributes
Sum of interior angles
What the equation gives you. Everything else feeds it.
Number of sides
The number of sides, a whole number of at least 3. The slider runs from 3 to 20. Raises the angle sum by 180° for every extra side, a straight-line rise. Raising it by 1 (to 7) moves the result by +25 %, from 720 to 900 °.
Worked example with the values above
- Number of triangles: n − 2 = 6 − 2 = 4.
- Each triangle contributes 180°.
- S = 4 × 180° = 720°.
Assumptions and limits
- A simple polygon: sides do not cross.
- Flat (plane) geometry.
Common mistakes
- Using n × 180° or (n − 1) × 180°.
- Applying a single-angle formula to an irregular polygon.
Statistics and uncertainty
This is exact geometry, with no statistical content. In surveying, the angle sum is used as a check: if the measured interior angles of a plot do not add up to (n − 2) × 180°, the difference is the total measurement error, which is then shared out.
Where it is used
- Checking the angles of a surveyed plot or a faceted component.
- Starting point for the angle of a regular polygon.
Maths you need
- Triangle angle sum: the three angles add to 180°
- Polygon: a closed shape with straight sides