Interior Angles, Exterior Angles and the Sum | Polygon Worksheets - Free Printable

To solve the problem of finding the sum of the interior angles of each polygon in the image, we need to use a fundamental geometric formula:

Formula for the Sum of Interior Angles of a Polygon:

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

Step-by-Step Solution:

1. Identify the Number of Sides (\( n \)) for Each Polygon:
- Carefully count the number of sides for each polygon in the image.

2. Apply the Formula:
- For each polygon, substitute the value of \( n \) into the formula \( (n - 2) \times 180^\circ \).

3. Calculate the Sum of Interior Angles:
- Perform the calculation for each polygon.

4. Fill in the Table:
- Record the number of sides and the calculated sum of interior angles for each polygon.

Example Calculations:

#### Polygon 1: Hexagon (6 sides)
- Number of sides (\( n \)): 6
- Sum of Interior Angles:
\[
(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]

#### Polygon 2: Pentagon (5 sides)
- Number of sides (\( n \)): 5
- Sum of Interior Angles:
\[
(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

#### Polygon 3: Heptagon (7 sides)
- Number of sides (\( n \)): 7
- Sum of Interior Angles:
\[
(7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ
\]

#### Polygon 4: Quadrilateral (4 sides)
- Number of sides (\( n \)): 4
- Sum of Interior Angles:
\[
(4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]

#### Polygon 5: Octagon (8 sides)
- Number of sides (\( n \)): 8
- Sum of Interior Angles:
\[
(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ
\]

#### Polygon 6: Triangle (3 sides)
- Number of sides (\( n \)): 3
- Sum of Interior Angles:
\[
(3 - 2) \times 180^\circ = 1 \times 180^\circ = 180^\circ
\]

General Approach for All Polygons:

- Repeat the above steps for each polygon in the image.
- Count the sides, apply the formula, and calculate the sum of interior angles.

Final Answer:

After completing the calculations for all polygons in the image, fill in the table with the number of sides and the corresponding sum of interior angles. The final answer will be a completed table with all values filled in.

\[
\boxed{\text{Completed Table with Number of Sides and Sum of Interior Angles}}
\]

If you provide the specific polygons or their side counts from the image, I can help fill in the exact values for you!