How to Solve: Polygon Problems

How to Solve: Polygon Problems

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

Introduction

"If you can solve polygon problems, you can design a soccer field, tile a bathroom floor, or even calculate how much fencing a farmer needs—all while crushing your geometry exam!

What You Need To Know First

Before tackling polygon problems, you must already understand: 1. Basic angle properties – Complementary, supplementary, and vertically opposite angles. 2. Triangle properties – Sum of interior angles = 180°, exterior angle theorem. 3. Regular vs. irregular polygons – A regular polygon has equal sides and equal angles; an irregular polygon does not.

If any of these are shaky, review them first—polygons build on these ideas!

Key Vocabulary

Formulas To Know

1. Sum of Interior Angles

Formula: Sum of interior angles = (n – 2) × 180° - n = number of sides (or vertices) - MEMORISE THIS – It’s the foundation for all polygon angle problems.

Example: For a hexagon (n = 6): Sum = (6 – 2) × 180° = 4 × 180° = 720°

2. Measure of One Interior Angle (Regular Polygon Only)

Formula: One interior angle = (n – 2) × 180° / n - n = number of sides - MEMORISE THIS – Only works for regular polygons (equal sides and angles).

Example: For a regular octagon (n = 8): One interior angle = (8 – 2) × 180° / 8 = 6 × 180° / 8 = 135°

3. Sum of Exterior Angles

Formula: Sum of exterior angles = 360° (always, for any convex polygon) - MEMORISE THIS – True for triangles, quadrilaterals, pentagons, etc.

Example: For a regular pentagon, each exterior angle = 360° / 5 = 72°

4. Measure of One Exterior Angle (Regular Polygon Only)

Formula: One exterior angle = 360° / n - n = number of sides - MEMORISE THIS – Only for regular polygons.

Example: For a regular decagon (n = 10): One exterior angle = 360° / 10 = 36°

5. Number of Diagonals

Formula: Number of diagonals = n(n – 3) / 2 - n = number of sides - Given on exam sheet (but useful to know).

Example: For a hexagon (n = 6): Diagonals = 6(6 – 3) / 2 = 6 × 3 / 2 = 9 diagonals

Step-by-Step Method

Follow these steps in order for any polygon problem:

1. Identify the polygon type – Is it regular or irregular? Convex or concave?
2. Count the sides (n) – Write down n clearly.
3. Determine what’s being asked – Interior angles? Exterior angles? Diagonals?
4. Pick the right formula – Use the list above.
5. Plug in n and solve – Show all working.
6. Check units and reasonableness – Angles should add up correctly; diagonals can’t be negative.

Worked Example Using the Steps

Problem: A regular polygon has an interior angle of 156°. How many sides does it have?

Step 1: Identify the polygon type → Regular (equal sides and angles). Step 2: Count sides (n) → Unknown (this is what we’re solving for). Step 3: What’s being asked? → Find n (number of sides). Step 4: Pick the formula → One interior angle = (n – 2) × 180° / n Step 5: Plug in and solve:
- 156° = (n – 2) × 180° / n
- 156n = 180(n – 2)
- 156n = 180n – 360
- 156n – 180n = –360
- –24n = –360
- n = 15 Step 6: Check → For n = 15, interior angle = (15 – 2) × 180° / 15 = 13 × 12° = 156° ✔️

Answer: The polygon has 15 sides.

Worked Examples

Example 1 – Basic (Sum of Interior Angles)

Problem: Find the sum of the interior angles of a nonagon (9-sided polygon).

Solution: 1. n = 9 (nonagon). 2. Use sum of interior angles formula: (n – 2) × 180° 3. Sum = (9 – 2) × 180° = 7 × 180° = 1260°

What we did and why: We used the formula for the sum of interior angles because the problem asked for the total of all angles, not just one. The number of sides (n = 9) was given directly.

Example 2 – Medium (Missing Angle in Irregular Polygon)

Problem: A quadrilateral has angles of 80°, 110°, and 95°. Find the fourth angle.

Solution: 1. n = 4 (quadrilateral). 2. Sum of interior angles = (4 – 2) × 180° = 360°. 3. Sum of known angles = 80° + 110° + 95° = 285°. 4. Fourth angle = 360° – 285° = 75°.

What we did and why: We used the sum formula to find the total, then subtracted the known angles. This works for any quadrilateral, regular or irregular.

Example 3 – Exam Style (Exterior Angles)

Problem: The exterior angles of a convex polygon are in the ratio 2:3:4:5:6. Find the measure of the smallest interior angle.

Solution: 1. Sum of exterior angles = 360° (always). 2. Let the exterior angles be 2x, 3x, 4x, 5x, 6x. 3. 2x + 3x + 4x + 5x + 6x = 360° → 20x = 360° → x = 18°. 4. Exterior angles: 36°, 54°, 72°, 90°, 108°. 5. Smallest exterior angle = 36°. 6. Smallest interior angle = 180° – 36° = 144° (since interior + exterior = 180°).

What we did and why: We used the fact that exterior angles sum to 360° and set up a ratio. Then, we found the smallest interior angle by subtracting the smallest exterior angle from 180°.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in—tonight, before your exam, here’s what you need to remember:

1. Sum of interior angles = (n – 2) × 180°. This works for any polygon, regular or irregular.
2. One interior angle (regular only) = (n – 2) × 180° / n. If the polygon isn’t regular, you can’t use this!
3. Exterior angles always add to 360°—no exceptions for convex polygons.
4. One exterior angle (regular only) = 360° / n. Super useful for finding n quickly.
5. Diagonals = n(n – 3) / 2. Don’t forget to divide by 2!

If you see a problem with angles in a ratio, set up an equation with the sum (360° for exterior, (n – 2) × 180° for interior). If it’s irregular, subtract known angles from the total. And always double-check: does the answer make sense? A 10-sided polygon can’t have 200° interior angles!

You’ve got this—go ace that exam!