How to Solve: Exterior Angles of Polygons

How to Solve: Exterior Angles of Polygons

Introduction

"If you can find the missing angle on a stop sign, you can solve any exterior angle problem on your exam—let’s break it down in 5 minutes."

What You Need To Know First

Before diving into exterior angles, make sure you understand: 1. Sum of interior angles of a polygon – The formula (n – 2) × 180° where n = number of sides. 2. Linear pairs – Adjacent angles on a straight line add up to 180°. 3. Regular vs. irregular polygons – Regular polygons have equal sides and equal angles; irregular polygons do not.

Key Vocabulary

Formulas To Know

Step-by-Step Method

How to solve any exterior angle problem in 4 steps:

1. Identify the polygon type
2. Is it regular (all sides/angles equal) or irregular (not equal)?

3. Count the number of sides (n).

4. Check what’s given

5. One exterior angle? Multiple angles? A missing angle?

6. If it’s a regular polygon, use 360° ÷ n to find each exterior angle.

7. Use the sum of exterior angles (360°)

8. If you know n – 1 exterior angles, subtract their sum from 360° to find the missing one.

9. If you know one exterior angle in a regular polygon, multiply it by n to check if it equals 360°.

10. Find interior angles if needed

11. Use Interior + Exterior = 180° to switch between them.

Worked Example Using the Steps

Problem: A regular polygon has an exterior angle of 40°. How many sides does it have?

Step 1: Identify the polygon type → Regular (given). Step 2: Given → Exterior angle = 40°. Step 3: Use Exterior angle = 360° ÷ n → 40° = 360° ÷ n. Step 4: Solve for n → n = 360° ÷ 40° = 9.

Answer: The polygon has 9 sides.

Worked Examples

Example 1 – Basic (Regular Polygon)

Problem: Find the exterior angle of a regular hexagon.

Solution: 1. A hexagon has n = 6 sides. 2. It’s regular → use Exterior angle = 360° ÷ n. 3. 360° ÷ 6 = 60°.

Answer: Each exterior angle is 60°.

What we did and why: - We used the formula for regular polygons because all exterior angles are equal. - The sum of all exterior angles is always 360°, so dividing by n gives one angle.

Example 2 – Medium (Irregular Polygon)

Problem: The exterior angles of a pentagon are 70°, 80°, 90°, and 100°. Find the missing exterior angle.

Solution: 1. A pentagon has n = 5 sides → sum of exterior angles = 360°. 2. Sum of given angles = 70° + 80° + 90° + 100° = 340°. 3. Missing angle = 360° – 340° = 20°.

Answer: The missing exterior angle is 20°.

What we did and why: - We used the sum of exterior angles (360°) because it works for any polygon. - Subtracted the known angles from 360° to find the missing one.

Example 3 – Exam Style (Disguised Problem)

Problem: A regular polygon has an interior angle of 140°. How many sides does it have?

Solution: 1. Find the exterior angle first (since we know the sum is 360°).
- Interior + Exterior = 180° → 140° + Exterior = 180° → Exterior = 40°. 2. Use the regular polygon formula:
- Exterior angle = 360° ÷ n → 40° = 360° ÷ n. 3. Solve for n:
- n = 360° ÷ 40° = 9.

Answer: The polygon has 9 sides.

What we did and why: - The problem gave an interior angle, but we needed the exterior angle to use the 360° rule. - We converted interior → exterior using the linear pair rule, then used the regular polygon formula.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in—30 seconds to exam success!

1. Exterior angles always add up to 360°, no matter the shape. That’s your golden rule.
2. For regular polygons, each exterior angle is 360° ÷ number of sides.
3. If it’s irregular, add up the given angles and subtract from 360° to find the missing one.
4. Interior + Exterior = 180°—use this to switch between them if the problem gives you one but asks for the other.
5. Watch out for traps: Is the polygon regular? Did you count all sides? Did you mix up interior and exterior?

Tonight, draw a triangle, quadrilateral, and pentagon. Label their exterior angles and check that they add to 360°. That’s it—you’ve got this!