Geometry Worksheets | Polygons Worksheets - Free Printable

To solve the problem, we need to determine the measure of one interior angle, one exterior angle, and the interior angle sum for each polygon. Let's go through the steps systematically.

Key Formulas:

1. Interior Angle Sum:
\[
\text{Interior Angle Sum} = (n - 2) \times 180^\circ
\]
where \( n \) is the number of sides of the polygon.

2. Measure of One Interior Angle:
\[
\text{Interior Angle} = \frac{\text{Interior Angle Sum}}{n}
\]

3. Measure of One Exterior Angle:
\[
\text{Exterior Angle} = \frac{360^\circ}{n}
\]

Step-by-Step Solution:

#### 1. Identify the number of sides (\( n \)) for each polygon:
- Regular Octagon: \( n = 8 \)
- Regular Hexagon: \( n = 6 \)
- Regular Pentagon: \( n = 5 \)

#### 2. Calculate the Interior Angle Sum, Interior Angle, and Exterior Angle for each polygon:

##### Polygon 1: Regular Octagon (\( n = 8 \))
- Interior Angle Sum:
\[
(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ
\]
- Interior Angle:
\[
\frac{1080^\circ}{8} = 135^\circ
\]
- Exterior Angle:
\[
\frac{360^\circ}{8} = 45^\circ
\]

##### Polygon 2: Regular Octagon (\( n = 8 \))
- Same as Polygon 1:
- Interior Angle Sum: \( 1080^\circ \)
- Interior Angle: \( 135^\circ \)
- Exterior Angle: \( 45^\circ \)

##### Polygon 3: Regular Octagon (\( n = 8 \))
- Same as Polygon 1:
- Interior Angle Sum: \( 1080^\circ \)
- Interior Angle: \( 135^\circ \)
- Exterior Angle: \( 45^\circ \)

##### Polygon 4: Regular Hexagon (\( n = 6 \))
- Interior Angle Sum:
\[
(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
- Interior Angle:
\[
\frac{720^\circ}{6} = 120^\circ
\]
- Exterior Angle:
\[
\frac{360^\circ}{6} = 60^\circ
\]

##### Polygon 5: Regular Octagon (\( n = 8 \))
- Same as Polygon 1:
- Interior Angle Sum: \( 1080^\circ \)
- Interior Angle: \( 135^\circ \)
- Exterior Angle: \( 45^\circ \)

##### Polygon 6: Regular Pentagon (\( n = 5 \))
- Interior Angle Sum:
\[
(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Interior Angle:
\[
\frac{540^\circ}{5} = 108^\circ
\]
- Exterior Angle:
\[
\frac{360^\circ}{5} = 72^\circ
\]

##### Polygon 7: Regular Pentagon (\( n = 5 \))
- Same as Polygon 6:
- Interior Angle Sum: \( 540^\circ \)
- Interior Angle: \( 108^\circ \)
- Exterior Angle: \( 72^\circ \)

##### Polygon 8: Regular Octagon (\( n = 8 \))
- Same as Polygon 1:
- Interior Angle Sum: \( 1080^\circ \)
- Interior Angle: \( 135^\circ \)
- Exterior Angle: \( 45^\circ \)

##### Polygon 9: Regular Octagon (\( n = 8 \))
- Same as Polygon 1:
- Interior Angle Sum: \( 1080^\circ \)
- Interior Angle: \( 135^\circ \)
- Exterior Angle: \( 45^\circ \)

##### Polygon 10: Regular Octagon (\( n = 8 \))
- Same as Polygon 1:
- Interior Angle Sum: \( 1080^\circ \)
- Interior Angle: \( 135^\circ \)
- Exterior Angle: \( 45^\circ \)

##### Polygon 11: Regular Octagon (\( n = 8 \))
- Same as Polygon 1:
- Interior Angle Sum: \( 1080^\circ \)
- Interior Angle: \( 135^\circ \)
- Exterior Angle: \( 45^\circ \)

##### Polygon 12: Regular Hexagon (\( n = 6 \))
- Same as Polygon 4:
- Interior Angle Sum: \( 720^\circ \)
- Interior Angle: \( 120^\circ \)
- Exterior Angle: \( 60^\circ \)

Final Answers:

\[
\boxed{
\begin{array}{|c|c|c|c|}
\hline
\text{Polygon} & \text{Interior Angle} & \text{Exterior Angle} & \text{Interior Angle Sum} \\
\hline
1 & 135^\circ & 45^\circ & 1080^\circ \\
2 & 135^\circ & 45^\circ & 1080^\circ \\
3 & 135^\circ & 45^\circ & 1080^\circ \\
4 & 120^\circ & 60^\circ & 720^\circ \\
5 & 135^\circ & 45^\circ & 1080^\circ \\
6 & 108^\circ & 72^\circ & 540^\circ \\
7 & 108^\circ & 72^\circ & 540^\circ \\
8 & 135^\circ & 45^\circ & 1080^\circ \\
9 & 135^\circ & 45^\circ & 1080^\circ \\
10 & 135^\circ & 45^\circ & 1080^\circ \\
11 & 135^\circ & 45^\circ & 1080^\circ \\
12 & 120^\circ & 60^\circ & 720^\circ \\
\hline
\end{array}
}
\]