Exterior Angle Calculator
Calculate exterior angles and explore the 360° sum theorem
Input Values
Results
Step-by-Step Solution
Formula: Exterior angle = 360° / n
Exterior angle = 360° / 6
Exterior angle = 0.00°
Formula: Interior = 180° - Exterior
Interior = 180° - 0.00°
Interior = 0.00°
For any polygon, the sum of exterior angles = 360°
Verification: 0.00 × 0.00° = 0.00° ≈ 360° ✓
Exterior Angles Reference Table
| Sides | Name | Exterior Angle | Interior Angle | Sum Check |
|---|---|---|---|---|
| 3 | Triangle | 120.00° | 60.00° | 60.00° + 120.00° = 180° |
| 4 | Quadrilateral | 90.00° | 90.00° | 90.00° + 90.00° = 180° |
| 5 | Pentagon | 72.00° | 108.00° | 108.00° + 72.00° = 180° |
| 6 | Hexagon | 60.00° | 120.00° | 120.00° + 60.00° = 180° |
| 7 | Heptagon | 51.43° | 128.57° | 128.57° + 51.43° = 180° |
| 8 | Octagon | 45.00° | 135.00° | 135.00° + 45.00° = 180° |
| 9 | Nonagon | 40.00° | 140.00° | 140.00° + 40.00° = 180° |
| 10 | Decagon | 36.00° | 144.00° | 144.00° + 36.00° = 180° |
| 11 | Hendecagon | 32.73° | 147.27° | 147.27° + 32.73° = 180° |
| 12 | Dodecagon | 30.00° | 150.00° | 150.00° + 30.00° = 180° |
Understanding Exterior Angles
What are Exterior Angles?
An exterior angle is formed when one side of a polygon is extended beyond the vertex. At each vertex, the exterior angle and the interior angle form a linear pair, meaning they are supplementary (sum to 180°). The exterior angle measures how much you "turn" when walking around the polygon.
The 360° Theorem
One of the most beautiful theorems in geometry: the sum of exterior angles of any polygon is always 360°, regardless of the number of sides or whether it's regular or irregular.
Why 360°?
Imagine walking around a polygon. At each vertex, you turn through the exterior angle. After walking completely around and returning to your starting point, you've made one full rotation of 360°. This works for any polygon!
Key Formulas
Each Exterior Angle (Regular Polygon): Exterior = 360° / n
Relationship to Interior: Exterior = 180° - Interior
Sum of All Exterior Angles: Always 360° (any polygon)
Finding Sides from Exterior: n = 360° / Exterior angle
Proof of 360° Sum
At each vertex, Interior + Exterior = 180°. For n vertices:
- Sum of (Interior + Exterior) at all vertices = n × 180°
- Sum of all interior angles = (n-2) × 180°
- Therefore: Sum of all exterior angles = n × 180° - (n-2) × 180°
- = n × 180° - n × 180° + 2 × 180°
- = 360° (for any value of n!)
Frequently Asked Questions
Why do exterior angles sum to 360°?
When you walk around a polygon and return to your starting point, you make one complete turn (360°). The exterior angles represent the amount you turn at each vertex, so they must sum to one full rotation.
Does this work for irregular polygons?
Yes! The 360° sum holds for all polygons, regular or irregular, convex or concave. This is one of the fundamental theorems of polygon geometry that applies universally.
How do you find an exterior angle?
For a regular polygon, divide 360° by the number of sides. Alternatively, subtract the interior angle from 180° (since they're supplementary). For irregular polygons, measure or calculate each exterior angle individually.
Can an exterior angle be negative?
In convex polygons, exterior angles are positive (0° to 180°). In concave polygons with reflex interior angles (>180°), the corresponding exterior angles can be considered negative, but the 360° sum still holds when accounting for direction.
What if I know the exterior angle but not the sides?
Use the formula n = 360° / exterior angle. For example, if each exterior angle is 72°, then n = 360° / 72° = 5 sides (a pentagon). This only works for regular polygons where all exterior angles are equal.
How are exterior angles used in navigation?
Exterior angles represent turns or changes in direction. In robotics and navigation, programming a robot to trace a polygon involves turning through the exterior angle at each vertex. The 360° sum ensures you return to your starting direction.
What's the largest exterior angle possible?
In a convex regular polygon, as the number of sides decreases, exterior angles increase. An equilateral triangle has the largest regular exterior angles at 120°. The theoretical maximum is 180° (which would be a straight line, not a polygon).
Do exterior angles work for 3D shapes?
The concept extends to 3D polyhedra differently. For spherical polygons on a sphere's surface, the exterior angle sum depends on the polygon's area. The 360° rule is specific to planar (flat) polygons.