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

Problem Overview:

The task involves completing formulas for interior and exterior angles of regular polygons and filling in a table based on these formulas. Let's break it down step by step.

---

Step 1: Understanding the Formulas for Regular Polygons

#### 1. Sum of Exterior Angles
For any polygon (regular or irregular), the sum of the exterior angles is always:
\[
\text{Sum of exterior angles} = 360^\circ
\]

#### 2. Interior Angle + Exterior Angle
At each vertex of a polygon, the interior angle and the exterior angle are supplementary. Therefore:
\[
\text{Interior angle} + \text{Exterior angle} = 180^\circ
\]

#### 3. Exterior Angle
For a regular polygon with \( n \) sides, each exterior angle is:
\[
\text{Exterior angle} = \frac{360^\circ}{n}
\]

#### 4. Interior Angle
Using the relationship between interior and exterior angles:
\[
\text{Interior angle} = 180^\circ - \text{Exterior angle}
\]
Substituting the formula for the exterior angle:
\[
\text{Interior angle} = 180^\circ - \frac{360^\circ}{n}
\]

#### 5. Sum of Interior Angles
The sum of the interior angles of a polygon with \( n \) sides is given by:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]

---

Step 2: Filling in the Table

Now, we will use these formulas to fill in the table. The table has columns for:
- \( n \) (number of sides)
- Exterior Angle
- Interior Angle
- Sum of Interior Angles

#### Row 1: \( n = 3 \) (Triangle)
- Exterior Angle:
\[
\text{Exterior angle} = \frac{360^\circ}{3} = 120^\circ
\]
- Interior Angle:
\[
\text{Interior angle} = 180^\circ - 120^\circ = 60^\circ
\]
- Sum of Interior Angles: Given as \( 180^\circ \).

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

#### Row 3: Exterior Angle = \( 18^\circ \)
- Number of Sides (\( n \)):
\[
n = \frac{360^\circ}{\text{Exterior angle}} = \frac{360^\circ}{18^\circ} = 20
\]
- Interior Angle:
\[
\text{Interior angle} = 180^\circ - 18^\circ = 162^\circ
\]
- Sum of Interior Angles:
\[
\text{Sum of interior angles} = (20 - 2) \times 180^\circ = 18 \times 180^\circ = 3240^\circ
\]

#### Row 4: Interior Angle = \( 140^\circ \)
- Exterior Angle:
\[
\text{Exterior angle} = 180^\circ - 140^\circ = 40^\circ
\]
- Number of Sides (\( n \)):
\[
n = \frac{360^\circ}{\text{Exterior angle}} = \frac{360^\circ}{40^\circ} = 9
\]
- Sum of Interior Angles:
\[
\text{Sum of interior angles} = (9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ
\]

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

#### Row 6: Sum of Interior Angles = \( 540^\circ \)
- Number of Sides (\( n \)):
\[
(n - 2) \times 180^\circ = 540^\circ \implies n - 2 = \frac{540^\circ}{180^\circ} = 3 \implies n = 5
\]
- Exterior Angle:
\[
\text{Exterior angle} = \frac{360^\circ}{5} = 72^\circ
\]
- Interior Angle:
\[
\text{Interior angle} = 180^\circ - 72^\circ = 108^\circ
\]

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

#### Row 8: Sum of Interior Angles = \( 360^\circ \)
- Number of Sides (\( n \)):
\[
(n - 2) \times 180^\circ = 360^\circ \implies n - 2 = \frac{360^\circ}{180^\circ} = 2 \implies n = 4
\]
- Exterior Angle:
\[
\text{Exterior angle} = \frac{360^\circ}{4} = 90^\circ
\]
- Interior Angle:
\[
\text{Interior angle} = 180^\circ - 90^\circ = 90^\circ
\]

#### Row 9: Sum of Interior Angles = \( 1440^\circ \)
- Number of Sides (\( n \)):
\[
(n - 2) \times 180^\circ = 1440^\circ \implies n - 2 = \frac{1440^\circ}{180^\circ} = 8 \implies n = 10
\]
- Exterior Angle:
\[
\text{Exterior angle} = \frac{360^\circ}{10} = 36^\circ
\]
- Interior Angle:
\[
\text{Interior angle} = 180^\circ - 36^\circ = 144^\circ
\]

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

---

Final Answer: Filled Table

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

\boxed{
\begin{array}{|c|c|c|c|}
\hline
n & \text{Exterior Angle} & \text{Interior Angle} & \text{Sum of Interior Angles} \\
\hline
3 & 120^\circ & 60^\circ & 180^\circ \\
6 & 60^\circ & 120^\circ & 720^\circ \\
20 & 18^\circ & 162^\circ & 3240^\circ \\
9 & 40^\circ & 140^\circ & 1260^\circ \\
8 & 45^\circ & 135^\circ & 1080^\circ \\
5 & 72^\circ & 108^\circ & 540^\circ \\
12 & 30^\circ & 150^\circ & 1800^\circ \\
4 & 90^\circ & 90^\circ & 360^\circ \\
10 & 36^\circ & 144^\circ & 1440^\circ \\
15 & 24^\circ & 156^\circ & 2340^\circ \\
\hline
\end{array}
}