Exterior Angles of Polygons

Learn what exterior angles are, how to find them, and discover why they always sum to 360 degrees.

Intermediate25 minLesson

Definition

An exterior angle of a polygon is formed when one side of the polygon is extended. It is the angle between this extended side and the adjacent side of the polygon.

For any polygon:

An exterior angle and its adjacent interior angle are supplementary (they add up to $180°$ )
If the interior angle is $\alpha$ , the exterior angle is $180° - \alpha$

The Exterior Angle Sum Theorem:

\text{Sum of all exterior angles} = 360°

This is true for any convex polygon, regardless of the number of sides!

For a regular polygon with

n

sides:

\text{Each exterior angle} = \frac{360°}{n}

Try it now

What is an exterior angle of a polygon?

Worked Examples

A triangle has interior angles of $50°$ , $60°$ , and $70°$ . Find all three exterior angles.

Use the supplementary relationship

Exterior angle $= 180° -$ Interior angle → Formula identified

Find exterior angle at $50°$ vertex

$180° - 50° = 130°$

Find exterior angle at $60°$ vertex

$180° - 60° = 120°$

Find exterior angle at $70°$ vertex

$180° - 70° = 110°$

Verify the sum

$130° + 120° + 110° = 360°$ → Confirmed!

Answer: The exterior angles are $130°$ , $120°$ , and $110°$ , which sum to $360°$ .

Common Mistakes

Confusing interior and exterior angles

Why it's wrong: Students sometimes calculate interior angles when asked for exterior, or vice versa.

Correct: Remember: exterior angles are formed by extending a side OUTSIDE the polygon. Interior + Exterior = $180°$ at each vertex.

Thinking the exterior angle sum depends on the number of sides

Why it's wrong: Since interior angle sums change with the number of sides, students assume exterior sums do too.

Correct: The sum of exterior angles is ALWAYS $360°$ for any convex polygon, regardless of how many sides it has.

Forgetting to extend only one side at each vertex

Why it's wrong: At each vertex, you can extend either side, but you should only count one exterior angle per vertex.

Correct: Choose one side to extend at each vertex. Each vertex contributes exactly one exterior angle to the sum.

Interactive Visual

Polygon explorer

Number of sidesPentagon · 5

Each exterior angle (regular polygon)

$360^\circ \div 5 = 72^\circ$

Sum of the exterior angles

$5 \cdot 72^\circ = 360^\circ$

Interior + exterior angle at one vertex

$108^\circ + 72^\circ = 180^\circ$

Walk around the polygon: in total you turn a full 360°, whatever the number of sides.

Angle Explorer

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Practice Problems

16 problems

Problem 1 of 16

Easy

What is an exterior angle of a polygon?

Why It Matters

Exterior angles are incredibly useful in geometry and real-world applications:

Navigation: Pilots and sailors use exterior angles when calculating turns. A 90-degree turn is an exterior angle of a square route.
Architecture: Designing buildings with angular facades requires understanding exterior angles.
Robotics: Programming a robot to trace a polygon path uses exterior angle calculations.
Sports: The angles at which a billiard ball bounces relate to exterior angles.

The beautiful fact that exterior angles always sum to 360 degrees provides a powerful shortcut for solving many geometry problems!

Real World Applications

Robot Navigation

When programming a robot to trace a polygon path, it needs to turn at each vertex. The turn angle equals the exterior angle!

Example:

A robot tracing a regular pentagon turns $\frac{360°}{5} = 72°$ at each corner. After 5 turns, it has rotated $360°$ total and faces its original direction.

1Try It Yourself

A delivery robot needs to trace a regular octagon path around a building.

How many degrees should it turn at each corner?

Step 1: Write the mathematical expression

Calculate: $\frac{360°}{8}$

Satellite Dish Installation

Satellite installers use angle measurements to properly aim dishes. Understanding exterior angles helps calculate mounting positions.

Example:

If a satellite dish needs to point at an angle that creates a $150°$ interior angle with the roof, the exterior angle (from the ground reference) is $180° - 150° = 30°$ .

2Try It Yourself

A building has a pentagonal rooftop. A satellite dish at one vertex makes an interior angle of $108°$ with the roof edge.

What is the exterior angle at this vertex?

Step 1: Write the mathematical expression

Calculate: $180° - 108°$

Key Takeaways

1An exterior angle is formed by extending one side of a polygon past a vertex
2Interior angle + Exterior angle = $180°$ (supplementary)
3The sum of all exterior angles of any convex polygon equals $360°$
4For a regular $n$ -sided polygon: each exterior angle = $\frac{360°}{n}$
5To find the number of sides: $n = \frac{360°}{\text{exterior angle}}$

Frequently Asked Questions

Imagine walking around the polygon, turning at each vertex. Each turn is an exterior angle. When you return to your starting point facing the same direction, you have turned a full circle:

360°

Glossary

Exterior angle: The angle formed between one side of a polygon and the extension of an adjacent side, measured outside the polygon.
Interior angle: The angle inside a polygon at a vertex, formed by two adjacent sides.
Supplementary angles: Two angles that add up to $180°$ .
Convex polygon: A polygon where all interior angles are less than $180°$ and all vertices point outward.
Regular polygon: A polygon with all sides equal in length and all angles equal in measure.