https://file.aiquickdraw.com/mathbot/file/1727235777133vh6bxs7s.jpg

To solve this problem involving the regular polygon (a pentagon in the image), let's break it down step by step.

### Step 1: Formula for the Area \( A \)

The area of a regular polygon can be calculated using the formula:
\[
A = \frac{1}{2} \cdot P \cdot h
\]
Where:
- \( P \) is the perimeter of the polygon.
- \( h \) is the apothem (the perpendicular distance from the center to a side).

Since we are dealing with a regular pentagon, we can express the perimeter \( P \) as:
\[
P = n \cdot b
\]
Where:
- \( n \) is the number of sides of the polygon (which is 5 in this case).
- \( b \) is the length of one side of the polygon.

Therefore, for a regular pentagon, the area becomes:
\[
A = \frac{1}{2} \cdot (5 \cdot b) \cdot h
\]
Simplifying the formula:
\[
A = \frac{5}{2} \cdot b \cdot h
\]

### Step 2: Solve for the Apothem \( h \)

Now we can solve for the apothem \( h \) from the area formula. Rearranging the area equation to solve for \( h \):
\[
A = \frac{5}{2} \cdot b \cdot h
\]
\[
h = \frac{2A}{5b}
\]

### Final Answers:

- **Area formula**: \( A = \frac{5}{2} \cdot b \cdot h \)
- **Height formula**: \( h = \frac{2A}{5b} \)

Would you like any additional clarification or further details on this? Here are some related questions that may help deepen your understanding:

1. How would the formula for the area change if the polygon had more or fewer sides?
2. Can you derive the formula for the area of a hexagon using the same approach?
3. What happens to the apothem when the side length \( b \) increases?
4. How can we estimate the perimeter of an irregular polygon?
5. How is the apothem related to the radius of the circumscribed circle of the polygon?

### Tip: 
For regular polygons, the apothem acts like the height of an equivalent triangle formed between the center and two adjacent vertices, making it essential in determining the area.

Write a formula for the area of the regular polygon and solve for the height h.

Learn how to calculate the area of a regular pentagon using the side length and the apothem. This step-by-step guide includes formulas for both the area and the height (apothem), suitable for students in Grades 8-10.

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