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Teacher Guide: Exterior Angles of Polygons

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

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10 questions on Polygons. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define exterior angles and identify them in polygons
  • Apply the relationship between interior and exterior angles
  • Use the Exterior Angle Sum Theorem to solve problems
  • Calculate individual exterior angles of regular polygons
  • Determine the number of sides of a polygon given its exterior angle
Prerequisites
  • • Understanding of interior angles of polygons
  • • Knowledge of the interior angle sum formula: (n−2)×180°(n-2) \times 180°
  • • Familiarity with supplementary angles
  • • Basic understanding of regular polygons
Discussion Starters
  • 1. If you walk around a polygon and turn at each corner, why do you end up facing your original direction?
  • 2. How could a drone use exterior angles to fly a patrol route around a building?
  • 3. Why is it easier to use exterior angles than interior angles to find the number of sides of a regular polygon?
  • 4. What happens to exterior angles as a polygon gets more and more sides?
Common Misconceptions

Exterior angles get smaller for polygons with more sides, so the sum must get smaller

Remediation: Use concrete examples: triangle has three 120°120° angles (360°360° total), hexagon has six 60°60° angles (360°360° total). More angles, smaller each, same total!

The exterior angle is outside the polygon, so it must be bigger than the interior angle

Remediation: Show a triangle where interior angles are 50°50°, 60°60°, 70°70° and exterior angles are 130°130°, 120°120°, 110°110°. Then show a hexagon where interior angles are 120°120° but exterior angles are only 60°60°.

Differentiation Ideas

For Struggling Students:

  • • Provide polygon cutouts so students can physically extend sides
  • • Focus only on regular polygons initially
  • • Use a visual number line from 0°0° to 180°180° to show interior/exterior pairs

For On-Level Students:

  • • Calculate exterior angles for various regular and irregular polygons
  • • Work backwards from exterior angles to find number of sides
  • • Solve problems mixing interior and exterior angle information

For Advanced Students:

  • • Explore why the sum is 360°360° using the turning turtle argument
  • • Investigate exterior angles of concave polygons
  • • Create proofs relating exterior and interior angle sums
Standards Alignment
  • 8.G.A.5 (CCSS.MATH.CONTENT.8.G.A.5)

    Use informal arguments to establish facts about the angle sum and exterior angle of triangles

  • HSG.CO.C.10 (CCSS.MATH.CONTENT.HSG.CO.C.10)

    Prove theorems about triangles, including measures of interior and exterior angles

Lesson Resources
  • visualInteractive Polygon Explorer

    Students extend sides to see exterior angles form

  • activityWalking the Polygon

    Physical activity where students walk a polygon shape and track their turns

  • worksheetExterior Angle Calculations

    Practice problems ranging from finding single angles to determining number of sides

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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°180°)
  • If the interior angle is α\alpha, the exterior angle is 180°−α180° - \alpha
The Exterior Angle Sum Theorem:
Sum of all exterior angles=360°\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 nn sides:
Each exterior angle=360°n\text{Each exterior angle} = \frac{360°}{n}

Worked Examples

A triangle has interior angles of 50°50°, 60°60°, and 70°70°. Find all three exterior angles.
1

Use the supplementary relationship

Exterior angle =180°−= 180° - Interior angle → Formula identified
2

Find exterior angle at 50°50° vertex

180°−50°=130°180° - 50° = 130°
3

Find exterior angle at 60°60° vertex

180°−60°=120°180° - 60° = 120°
4

Find exterior angle at 70°70° vertex

180°−70°=110°180° - 70° = 110°
5

Verify the sum

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

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°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°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.

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 360°5=72°\frac{360°}{5} = 72° at each corner. After 5 turns, it has rotated 360°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: 360°8\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°150° interior angle with the roof, the exterior angle (from the ground reference) is 180°−150°=30°180° - 150° = 30°.

2Try It Yourself

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

What is the exterior angle at this vertex?

Step 1: Write the mathematical expression

Calculate: 180°−108°180° - 108°

Key Takeaways

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

Frequently Asked Questions

Why do exterior angles always sum to 360 degrees?

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°360°!

Does this work for concave (non-convex) polygons?

The 360°360° rule applies to convex polygons. For concave polygons, some exterior angles are measured differently (they can be negative in some conventions), but the concept still applies with proper definitions.

What is the smallest exterior angle possible?

For a regular polygon, the exterior angle approaches 0°0° as the number of sides approaches infinity (like a circle). The largest is 120°120° for an equilateral triangle.

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°180°.
Convex polygon
A polygon where all interior angles are less than 180°180° and all vertices point outward.
Regular polygon
A polygon with all sides equal in length and all angles equal in measure.

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