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

Let's solve this step by step.

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Step 1: Complete the formulas

We are given a regular polygon with $ n $ sides. Let’s fill in the blanks for the formulas:

1. Sum of exterior angles = $ \boxed{360^\circ} $

> ✔ The sum of the exterior angles of any polygon is always $ 360^\circ $, regardless of the number of sides.

2. Interior angle + Exterior angle = $ \boxed{180^\circ} $

> ✔ These two angles form a straight line at each vertex.

3. Exterior angle = $ \boxed{360^\circ} \div n $

> ✔ Since the total sum of exterior angles is $ 360^\circ $, and all exterior angles are equal in a regular polygon, each one is $ \frac{360^\circ}{n} $.

4. Sum of interior angles = $ \boxed{180^\circ} \times (n - \boxed{2}) $

> ✔ This is the standard formula: $ (n - 2) \times 180^\circ $

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Now we can use these formulas to complete the table.

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Step 2: Fill in the Table

We'll go row by row and use the formulas.

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#### Row 1: $ n = 3 $
- Sum of interior angles = $ 180^\circ $ → Given
- So, $ (3 - 2) \times 180 = 180^\circ $ ✔
- Exterior angle = $ \frac{360^\circ}{3} = 120^\circ $
- Interior angle = $ 180^\circ - 120^\circ = 60^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 3 | 120° | 60° | 180° |

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#### Row 2: $ n = 6 $
- $ n = 6 $
- Exterior angle = $ \frac{360^\circ}{6} = 60^\circ $
- Interior angle = $ 180^\circ - 60^\circ = 120^\circ $
- Sum of interior angles = $ (6 - 2) \times 180 = 4 \times 180 = 720^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 6 | 60° | 120° | 720° |

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#### Row 3: Exterior angle = 18°
- $ \text{Exterior angle} = \frac{360^\circ}{n} = 18^\circ $
- So, $ n = \frac{360}{18} = 20 $
- Interior angle = $ 180^\circ - 18^\circ = 162^\circ $
- Sum of interior angles = $ (20 - 2) \times 180 = 18 \times 180 = 3240^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 20 | 18° | 162° | 3240° |

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#### Row 4: Interior angle = 140°
- $ \text{Interior angle} = 140^\circ $
- So, $ \text{Exterior angle} = 180^\circ - 140^\circ = 40^\circ $
- $ \text{Exterior angle} = \frac{360^\circ}{n} \Rightarrow n = \frac{360}{40} = 9 $
- Sum of interior angles = $ (9 - 2) \times 180 = 7 \times 180 = 1260^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 9 | 40° | 140° | 1260° |

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#### Row 5: $ n = 8 $
- $ n = 8 $
- Exterior angle = $ \frac{360}{8} = 45^\circ $
- Interior angle = $ 180 - 45 = 135^\circ $
- Sum of interior angles = $ (8 - 2) \times 180 = 6 \times 180 = 1080^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 8 | 45° | 135° | 1080° |

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#### Row 6: Sum of interior angles = 540°
- $ (n - 2) \times 180 = 540 $
- $ n - 2 = \frac{540}{180} = 3 $
- $ n = 5 $
- Exterior angle = $ \frac{360}{5} = 72^\circ $
- Interior angle = $ 180 - 72 = 108^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 5 | 72° | 108° | 540° |

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#### Row 7: Interior angle = 150°
- $ \text{Exterior angle} = 180 - 150 = 30^\circ $
- $ n = \frac{360}{30} = 12 $
- Sum of interior angles = $ (12 - 2) \times 180 = 10 \times 180 = 1800^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 12 | 30° | 150° | 1800° |

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#### Row 8: Sum of interior angles = 360°
- $ (n - 2) \times 180 = 360 $
- $ n - 2 = 2 \Rightarrow n = 4 $
- Exterior angle = $ \frac{360}{4} = 90^\circ $
- Interior angle = $ 180 - 90 = 90^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 4 | 90° | 90° | 360° |

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#### Row 9: Sum of interior angles = 1440°
- $ (n - 2) \times 180 = 1440 $
- $ n - 2 = \frac{1440}{180} = 8 $
- $ n = 10 $
- Exterior angle = $ \frac{360}{10} = 36^\circ $
- Interior angle = $ 180 - 36 = 144^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 10 | 36° | 144° | 1440° |

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#### Row 10: Interior angle = 156°
- $ \text{Exterior angle} = 180 - 156 = 24^\circ $
- $ n = \frac{360}{24} = 15 $
- Sum of interior angles = $ (15 - 2) \times 180 = 13 \times 180 = 2340^\circ $

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|------------------------|
| 15 | 24° | 156° | 2340° |

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✔ Final Completed Table

| $ 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° |

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✔ Final Answers for Formulas:

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

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✔ All filled correctly!