The polygon shown in the image is a regular octagon, and to find its area, we can use the following formula for the area of a regular polygon:

$A = \frac{1}{2} \times P \times a$

Where:

• $P$ is the perimeter of the polygon.
• $a$ is the apothem (the distance from the center to the midpoint of a side).

From the image, we have:

• The apothem $a = 6$ inches.
• The side length $s = 3.7$ inches.

Step 1: Find the Perimeter

The perimeter $P$ of a regular polygon is given by:

$P = n \times s$

For an octagon, $n = 8$. So,

$P = 8 \times 3.7 = 29.6 \, \text{inches}$

Step 2: Calculate the Area

Now, using the formula for the area:

$A = \frac{1}{2} \times 29.6 \times 6 = 88.8 \, \text{square inches}$

Thus, the area of the polygon is 88.8 square inches.

Do you need more details or have any questions?

Here are 5 related questions:

1. How would you find the area of a regular hexagon?
2. How do you calculate the perimeter of any polygon?
3. What is the apothem and how is it used in calculating the area of regular polygons?
4. What is the difference between the perimeter and the area of a shape?
5. How does changing the side length affect the area of a regular polygon?

Tip: For any regular polygon, knowing the apothem and the side length simplifies the area calculation significantly!