How to Solve: Interior Angles

How to Solve: Interior Angles

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Imagine you’re designing a hexagonal tile floor—how do you know if the angles will fit together perfectly? Mastering interior angles lets you solve that AND crush every polygon question on your exam!

What You Need To Know First

1. Basic angle properties (e.g., angles in a triangle sum to 180°).
2. Names of polygons (triangle, quadrilateral, pentagon, etc.).
3. Regular vs. irregular polygons (regular = all sides and angles equal).

Key Vocabulary

Formulas To Know

1. Sum of Interior Angles (for any polygon)

Formula: Sum = (n – 2) × 180° - n = number of sides (or vertices) in the polygon. - MEMORISE THIS – It’s not always on the exam sheet!

Example: For a pentagon (n = 5): Sum = (5 – 2) × 180° = 3 × 180° = 540°

2. Measure of One Interior Angle (for regular polygons only)

Formula: One interior angle = (n – 2) × 180° / n - n = number of sides. - MEMORISE THIS – Examiners love testing this!

Example: For a regular hexagon (n = 6): One angle = (6 – 2) × 180° / 6 = 4 × 180° / 6 = 120°

3. Relationship Between Interior and Exterior Angles

Formula: Interior angle + Exterior angle = 180° - Given on exam sheet (but you must know how to use it!).

Example: If an exterior angle = 30°, then the interior angle = 180° – 30° = 150°.

Step-by-Step Method

How to Find the Sum of Interior Angles

1. Count the sides of the polygon. Call this number n.
2. Plug into the formula: Sum = (n – 2) × 180°.
3. Calculate the result.

Example: Find the sum of interior angles in an octagon. 1. n = 8 (octagon has 8 sides). 2. Sum = (8 – 2) × 180° = 6 × 180°. 3. Sum = 1080°.

How to Find One Interior Angle (Regular Polygon Only)

1. Count the sides (n).
2. Use the formula: One angle = (n – 2) × 180° / n.
3. Calculate the result.

Example: Find one interior angle of a regular decagon (10 sides). 1. n = 10. 2. One angle = (10 – 2) × 180° / 10 = 8 × 180° / 10. 3. One angle = 144°.

How to Find a Missing Interior Angle (Irregular Polygon)

1. Find the sum of interior angles using (n – 2) × 180°.
2. Add up the known angles.
3. Subtract the sum of known angles from the total sum to find the missing angle.

Example: A quadrilateral has angles 80°, 100°, and 95°. Find the missing angle. 1. Sum = (4 – 2) × 180° = 360°. 2. Known angles = 80° + 100° + 95° = 275°. 3. Missing angle = 360° – 275° = 85°.

Worked Examples

Example 1 – Basic (Sum of Angles)

Find the sum of interior angles in a heptagon (7 sides). 1. n = 7. 2. Sum = (7 – 2) × 180° = 5 × 180°. 3. Sum = 900°.

What we did and why: We used the formula for the sum of interior angles because we only needed the total, not individual angles.

Example 2 – Medium (One Angle in Regular Polygon)

Find one interior angle of a regular nonagon (9 sides). 1. n = 9. 2. One angle = (9 – 2) × 180° / 9 = 7 × 180° / 9. 3. One angle = 140°.

What we did and why: We used the regular polygon formula because all angles are equal in a regular nonagon.

Example 3 – Exam Style (Missing Angle in Irregular Polygon)

A pentagon has angles 100°, 120°, 110°, and 95°. Find the missing angle. 1. Sum = (5 – 2) × 180° = 540°. 2. Known angles = 100° + 120° + 110° + 95° = 425°. 3. Missing angle = 540° – 425° = 115°.

What we did and why: We found the total sum first, then subtracted the known angles to isolate the missing one.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in—night before the exam style!

1. Sum of interior angles? Use (n – 2) × 180°. That’s your go-to for any polygon.
2. One angle in a regular polygon? Take the sum, then divide by n.
3. Missing angle in an irregular polygon? Find the total sum, add up the known angles, then subtract.
4. Watch out for traps! If the question doesn’t say ‘regular,’ don’t assume it is. And if they give you an exterior angle, remember: interior + exterior = 180°.

Now go crush that exam—you’ve got this!