Exterior Angles of a Polygon: Sum & Formula
Exterior Angles of a Polygon: Sum & Formula
TL;DR
The exterior angles of a polygon are the angles between each side and the extension of its neighbour, and they always sum to 360°. In a regular polygon, each exterior angle is ( \frac{360°}{n} ). This article defines the exterior angle, derives the 360° sum, gives the formula, and works through examples.
What Is an Exterior Angle of a Polygon?
An exterior angle of a polygon is the angle formed outside the polygon between one side and the extension of the side next to it. Each vertex has an exterior angle that pairs with the interior angle at the same vertex.
Because the side and its extension form a straight line, together they make a straight angle:
[ \text{interior angle} + \text{exterior angle} = 180°. ]
So the exterior angle is the supplement of the interior angle at the same vertex.
Why the Exterior Angles Always Sum to 360°
This is the result everything else depends on. Take any polygon with ( n ) sides. Each vertex contributes one interior and one exterior angle, and the two add to ( n \times 180°. ) Now subtract the sum of the interior angles for an ( n )-sided polygon:
[ \text{sum of exterior angles} = n \times 180° - (n - 2) \times 180° = 360° .]
The 180°n terms cancel, and 360° is all that remains. This holds true for regular and irregular polygons alike.
The Exterior Angle of a Regular Polygon
In a regular polygon, each exterior angle is ( \frac{360°}{n} ).
For example:
- A regular hexagon has ( n=6 ) and each exterior angle is ( \frac{360°}{6} = 60° ).
- A square has ( n=4 ) and each exterior angle is ( 90° ).
| Regular polygon | Sides (n) | Each exterior angle (( \frac{360°}{n} )) |
|---|---|---|
| Equilateral triangle | 3 | 120° |
| Square | 4 | 90° |
| Regular pentagon | 5 | 72° |
| Regular hexagon | 6 | 60° |
| Regular octagon | 8 | 45° |
| Regular decagon | 10 | 36° |
Examples of Exterior Angles of a Polygon
Example 1
Find each exterior angle of a regular octagon.
Each exterior angle = ( \frac{360°}{8} = 45° ).
Example 2
Find each interior angle of a regular pentagon using its exterior angle.
Each exterior angle = ( \frac{360°}{5} = 72° ). The interior angle is its supplement: ( 180° - 72° = 108° ).
Example 3
Three exterior angles of a quadrilateral are 80°, 95°, and 70°. Find the fourth.
The fourth angle is ( 360° - (80° + 95° + 70°) = 115° ).
Example 4
A regular polygon has each exterior angle equal to 40°. How many sides does it have?
Use ( n = \frac{360°}{40°} = 9 ).
Example 5
A regular polygon has each interior angle 150°.
Each exterior angle is ( 180° - 150° = 30° ). Then ( n = \frac{360°}{30°} = 12 ).
Example 6
Can a regular polygon have an exterior angle of 50°?
Using ( n = \frac{360°}{50°} = 7.2 ), it must be a whole number, so no.
Why Exterior Angles Matter Beyond the Classroom
The constant 360° turn is an important rule that anything following a closed path obeys, with applications in:
- Robotics
- Surveying
- Road design
- Tiling
Key Takeaways
- The exterior angles of a polygon always sum to 360°.
- For a regular polygon, each exterior angle is ( \frac{360°}{n} ).
- The most common mistake is mixing totals between interior and exterior angles.
Practice Problems
- Find each exterior angle of a regular decagon.
- A regular polygon has each exterior angle 24°. How many sides does it have?
- Four exterior angles of a pentagon are 60°, 75°, 80°, and 65°. Find the fifth.