Exterior Angles of Polygons
Learn what exterior angles are, how to find them, and discover why they always sum to 360 degrees.
Definition
- An exterior angle and its adjacent interior angle are supplementary (they add up to )
- If the interior angle is , the exterior angle is
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Worked Examples
A triangle has interior angles of , , and . Find all three exterior angles.
Use the supplementary relationship
Exterior angle Interior angle → Formula identified
Find exterior angle at vertex
→
Find exterior angle at vertex
→
Find exterior angle at vertex
→
Verify the sum
→ Confirmed!
Answer: The exterior angles are , , and , which sum to .
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 = 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 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.
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Practice Problems
16 problemsWhat is an exterior angle of a polygon?
Why It Matters
- 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.
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 at each corner. After 5 turns, it has rotated total and faces its original direction.
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:
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 interior angle with the roof, the exterior angle (from the ground reference) is .
A building has a pentagonal rooftop. A satellite dish at one vertex makes an interior angle of with the roof edge.
What is the exterior angle at this vertex?
Step 1: Write the mathematical expression
Calculate:
Key Takeaways
- 1An exterior angle is formed by extending one side of a polygon past a vertex
- 2Interior angle + Exterior angle = (supplementary)
- 3The sum of all exterior angles of any convex polygon equals
- 4For a regular -sided polygon: each exterior angle =
- 5To find the number of sides:
Frequently Asked Questions
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 .
- Convex polygon
- A polygon where all interior angles are less than and all vertices point outward.
- Regular polygon
- A polygon with all sides equal in length and all angles equal in measure.