Angles in Regular Polygons Worksheet | Fun and Engaging Geometry ... - Free Printable

Let’s solve this step by step.

We are working with regular polygons — shapes where all sides and angles are equal.

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Step 1: Fill in the formulas

For any regular polygon with n sides:

- Sum of exterior angles = always 360° (this is true for ANY convex polygon, regular or not)

- Interior angle + Exterior angle = 180° (they form a straight line at each vertex)

- Exterior angle = 360° ÷ n

- Sum of interior angles = 180° × (n – 2)

So we fill in the formula boxes:

> Sum of exterior angles = 360°
> Interior angle + Exterior angle = 180°
> Exterior angle = 360 ÷ n
> Sum of interior angles = 180 x (n – 2)

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Step 2: Use these formulas to complete the table

We’ll go row by row.

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#### Row 1: n = 3 (triangle)

- Exterior angle = 360 ÷ 3 = 120°
- Interior angle = 180 – 120 = 60°
- Sum of interior angles = given as 180° → check: 180×(3–2)=180 ✔️

→ Fill: Exterior = 120°, Interior = 60°

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#### Row 2: n = 6 (hexagon)

- Exterior angle = 360 ÷ 6 = 60°
- Interior angle = 180 – 60 = 120°
- Sum of interior angles = 180 × (6 – 2) = 180 × 4 = 720°

→ Fill: Exterior = 60°, Interior = 120°, Sum = 720°

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#### Row 3: Exterior angle = 18°

Find n:
n = 360 ÷ 18 = 20

Then:
- Interior angle = 180 – 18 = 162°
- Sum of interior angles = 180 × (20 – 2) = 180 × 18 = 3240°

→ Fill: n = 20, Interior = 162°, Sum = 3240°

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#### Row 4: Interior angle = 140°

Then exterior angle = 180 – 140 = 40°

n = 360 ÷ 40 = 9

Sum of interior angles = 180 × (9 – 2) = 180 × 7 = 1260°

→ Fill: n = 9, Exterior = 40°, Sum = 1260°

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#### Row 5: n = 8 (octagon)

- Exterior angle = 360 ÷ 8 = 45°
- Interior angle = 180 – 45 = 135°
- Sum of interior angles = 180 × (8 – 2) = 180 × 6 = 1080°

→ Fill: Exterior = 45°, Interior = 135°, Sum = 1080°

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#### Row 6: Sum of interior angles = 540°

Use formula:
180 × (n – 2) = 540
→ n – 2 = 540 ÷ 180 = 3
→ n = 5

Then:
- Exterior angle = 360 ÷ 5 = 72°
- Interior angle = 180 – 72 = 108°

→ Fill: n = 5, Exterior = 72°, Interior = 108°

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#### Row 7: Interior angle = 150°

Exterior angle = 180 – 150 = 30°

n = 360 ÷ 30 = 12

Sum of interior angles = 180 × (12 – 2) = 180 × 10 = 1800°

→ Fill: n = 12, Exterior = 30°, Sum = 1800°

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#### Row 8: Sum of interior angles = 360°

180 × (n – 2) = 360
→ n – 2 = 2
→ n = 4 (square)

Exterior angle = 360 ÷ 4 = 90°

Interior angle = 180 – 90 = 90°

→ Fill: n = 4, Exterior = 90°, Interior = 90°

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#### Row 9: Sum of interior angles = 1440°

180 × (n – 2) = 1440
→ n – 2 = 1440 ÷ 180 = 8
→ n = 10

Exterior angle = 360 ÷ 10 = 36°

Interior angle = 180 – 36 = 144°

→ Fill: n = 10, Exterior = 36°, Interior = 144°

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#### Row 10: Interior angle = 156°

Exterior angle = 180 – 156 = 24°

n = 360 ÷ 24 = 15

Sum of interior angles = 180 × (15 – 2) = 180 × 13 = 2340°

→ Fill: n = 15, Exterior = 24°, Sum = 2340°

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Final Check: All values make sense?

Yes — every calculation follows from the core rules:

- Exterior angles sum to 360°
- Each interior + exterior = 180°
- Sum of interiors = 180(n–2)

All numbers are consistent.

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Final Answer:

Formulas:

Sum of exterior angles = 360°
Interior angle + Exterior angle = 180°
Exterior angle = 360 ÷ n
Sum of interior angles = 180 x (n – 2)

Table filled:

| n | Exterior Angle | Interior Angle | Sum of Interior Angles |
|-----|----------------|----------------|------------------------|
| 3 | 120° | 60° | 180° |
| 6 | 60° | 120° | 720° |
| 20 | 18° | 162° | 3240° |
| 9 | 40° | 140° | 1260° |
| 8 | 45° | 135° | 1080° |
| 5 | 72° | 108° | 540° |
| 12 | 30° | 150° | 1800° |
| 4 | 90° | 90° | 360° |
| 10 | 36° | 144° | 1440° |
| 15 | 24° | 156° | 2340° |