Interior and Exterior Angles in Regular Polygons - Free Printable

Let's solve the problem step by step for each shape in the worksheet titled "Interior and Exterior Angles".

We’ll analyze each polygon: triangle, square, pentagon, hexagon, heptagon (7 sides), and octagon (8 sides). For each, we will calculate:

- Sum of interior angles
- Size of each interior angle (if regular)
- Total exterior angle sum
- Size of each exterior angle (if regular)

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🔷 1. Triangle (3 sides)

Given:
- Each interior angle = 60°
- It’s an equilateral triangle, so all angles are equal.

#### ✔ Calculations:
- Sum of interior angles = $ (n - 2) \times 180^\circ = (3 - 2) \times 180 = 180^\circ $
- Size of each interior angle = $ \frac{180^\circ}{3} = 60^\circ $
- Total exterior angle = Always $ 360^\circ $ for any polygon
- Size of each exterior angle = $ \frac{360^\circ}{3} = 120^\circ $

> ✔️ Already filled: Sum of interior angles = 180°
> ✔️ Fill in:
> - Size of each interior angle: 60°
> - Total exterior angle: 360°
> - Size of each exterior angle: 120°

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🔷 2. Square (4 sides)

Regular quadrilateral (all sides and angles equal).

#### ✔ Calculations:
- Sum of interior angles = $ (4 - 2) \times 180 = 360^\circ $
- Size of each interior angle = $ \frac{360^\circ}{4} = 90^\circ $
- Total exterior angle = $ 360^\circ $
- Size of each exterior angle = $ \frac{360^\circ}{4} = 90^\circ $

> ✔️ Fill in:
> - Size of each interior angle: 90°
> - Total exterior angle: 360°
> - Size of each exterior angle: 90°

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🔷 3. Pentagon (5 sides)

Regular pentagon.

#### ✔ Calculations:
- Sum of interior angles = $ (5 - 2) \times 180 = 3 \times 180 = 540^\circ $
- Size of each interior angle = $ \frac{540^\circ}{5} = 108^\circ $
- Total exterior angle = $ 360^\circ $
- Size of each exterior angle = $ \frac{360^\circ}{5} = 72^\circ $

> ✔️ Fill in:
> - Size of each interior angle: 108°
> - Total exterior angle: 360°
> - Size of each exterior angle: 72°

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🔷 4. Hexagon (6 sides)

Regular hexagon.

#### ✔ Calculations:
- Sum of interior angles = $ (6 - 2) \times 180 = 4 \times 180 = 720^\circ $
- Size of each interior angle = $ \frac{720^\circ}{6} = 120^\circ $
- Total exterior angle = $ 360^\circ $
- Size of each exterior angle = $ \frac{360^\circ}{6} = 60^\circ $

> ✔️ Fill in:
> - Size of each interior angle: 120°
> - Total exterior angle: 360°
> - Size of each exterior angle: 60°

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🔷 5. Heptagon (7 sides)

Regular heptagon.

#### ✔ Calculations:
- Sum of interior angles = $ (7 - 2) \times 180 = 5 \times 180 = 900^\circ $
- Size of each interior angle = $ \frac{900^\circ}{7} \approx 128.57^\circ $
- Total exterior angle = $ 360^\circ $
- Size of each exterior angle = $ \frac{360^\circ}{7} \approx 51.43^\circ $

> ✔️ Fill in:
> - Size of each interior angle: ≈128.57°
> - Total exterior angle: 360°
> - Size of each exterior angle: ≈51.43°

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🔷 6. Octagon (8 sides)

Regular octagon.

#### ✔ Calculations:
- Sum of interior angles = $ (8 - 2) \times 180 = 6 \times 180 = 1080^\circ $
- Size of each interior angle = $ \frac{1080^\circ}{8} = 135^\circ $
- Total exterior angle = $ 360^\circ $
- Size of each exterior angle = $ \frac{360^\circ}{8} = 45^\circ $

> ✔️ Fill in:
> - Size of each interior angle: 135°
> - Total exterior angle: 360°
> - Size of each exterior angle: 45°

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## ✔ Final Answer Summary Table

| Shape | Interior Angle | Total Exterior | Exterior Angle |
|------------|----------------|----------------|----------------|
| Triangle | 60° | 360° | 120° |
| Square | 90° | 360° | 90° |
| Pentagon | 108° | 360° | 72° |
| Hexagon | 120° | 360° | 60° |
| Heptagon | ≈128.57° | 360° | ≈51.43° |
| Octagon | 135° | 360° | 45° |

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🔍 Key Concepts Used:

- Sum of interior angles of an *n*-gon: $ (n - 2) \times 180^\circ $
- Each interior angle (regular polygon): $ \frac{(n - 2) \times 180^\circ}{n} $
- Total exterior angles: Always $ 360^\circ $
- Each exterior angle (regular polygon): $ \frac{360^\circ}{n} $
- Interior + Exterior angle at a vertex = $ 180^\circ $

Let me know if you'd like this formatted as a printable worksheet or with diagrams!