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

Explanation:
Let’s fill in the formulas and table step by step.

Part 1: Formulas for regular polygons with n sides

1. Sum of exterior angles
For *any* polygon (regular or not), the sum of the exterior angles (one per vertex, taken in the same direction) is always:
→ 360°
So:
Sum of exterior angles = 360°

2. Interior angle + Exterior angle
At any vertex, an interior angle and its adjacent exterior angle form a straight line (they are supplementary).
So:
Interior angle + Exterior angle = 180°

3. Exterior angle
In a *regular* polygon, all exterior angles are equal. Since their sum is 360° and there are n of them:
Exterior angle = 360° ÷ n

4. Sum of interior angles
We know:
- Each interior angle = 180° − exterior angle = 180° − (360°/n)
- So sum = n × [180° − (360°/n)] = 180°n − 360° = 180° × (n − 2)
So:
Sum of interior angles = 180 × (n − 2)

Now let’s fill the table using these formulas.

We’ll go row by row.

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Row 1: n = 3 (triangle)
- Sum of interior angles = 180° (given) → matches 180×(3−2) = 180° ✔
- Exterior angle = 360° ÷ 3 = 120°
- Interior angle = 180° − 120° = 60°

→ 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°
We need to find n first:
Exterior = 360 ÷ n = 18 → 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 = 180 − 140 = 40°
n = 360 ÷ 40 = 9
Sum = 180 × (9 − 2) = 180 × 7 = 1260°

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

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Row 5: n = 8 (octagon)
- Exterior = 360 ÷ 8 = 45°
- Interior = 180 − 45 = 135°
- Sum = 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: Sum = 180(n − 2) = 540
→ n − 2 = 540 ÷ 180 = 3
→ n = 5
Then:
- Exterior = 360 ÷ 5 = 72°
- Interior = 180 − 72 = 108°

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

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Row 7: Interior angle = 150°
Exterior = 180 − 150 = 30°
n = 360 ÷ 30 = 12
Sum = 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
Exterior = 360 ÷ 4 = 90°
Interior = 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 = 360 ÷ 10 = 36°
Interior = 180 − 36 = 144°

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

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Row 10: Interior angle = 156°
Exterior = 180 − 156 = 24°
n = 360 ÷ 24 = 15
Sum = 180 × (15 − 2) = 180 × 13 = 2340°

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

Now double-check a few:
- n=5: sum=540°, interior=108°, exterior=72° → 5×108=540 ✔
- n=10: sum=1440°, interior=144°, 10×144=1440 ✔
- n=15: interior=156°, 15×156=2340 ✔

All consistent.

Final Answer:
Formulas:
- Sum of exterior angles = 360°
- Interior angle + Exterior angle = 180°
- Exterior angle = 360 ÷ n
- Sum of interior angles = 180 × (n − 2)

Table filled as:

| 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° |