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.