What is the formula for finding the area of a regular polygon with perimeter [tex]P[/tex] and apothem length [tex]a[/tex]?

A. [tex]A = \frac{1}{2} P a[/tex]
B. [tex]a = 2 P A[/tex]
C. [tex]A = 2 P a[/tex]
D. [tex]a = \frac{1}{2} P A[/tex]



Answer :

To find the area [tex]\( A \)[/tex] of a regular polygon with perimeter [tex]\( P \)[/tex] and apothem length [tex]\( a \)[/tex], we use the specific formula designed for this scenario. Let's go through the steps to identify the correct formula:

1. Understand the Problem:
We need to determine the mathematical expression that correctly calculates the area of a regular polygon given its perimeter [tex]\( P \)[/tex] and apothem length [tex]\( a \)[/tex].

2. Recognize the Formula:
The area [tex]\( A \)[/tex] of a regular polygon can be calculated using the formula:
[tex]\[ A = \frac{1}{2} \times P \times a \][/tex]
This formula comes from the fact that a regular polygon can be divided into isosceles triangles. The apothem [tex]\( a \)[/tex] acts as the height of each triangle, and the perimeter [tex]\( P \)[/tex] when divided by the number of sides gives the base lengths of each triangle.

3. Match the Given Options:
Let's look at the options provided and match them against the recognized formula:

- Option A: [tex]\( A = \frac{1}{2} (P \times a) \)[/tex]

This matches our recognized formula exactly.

- Option B: [tex]\( a = 2 P A \)[/tex]

This does not represent the correct relationship for finding the area.

- Option C: [tex]\( A = 2 P a \)[/tex]

This overstates the formula by a factor of 4.

- Option D: [tex]\( a = \frac{1}{2} (P \times A) \)[/tex]

This rearranges terms incorrectly and is not the right formula.

4. Conclusion:
The correct formula for finding the area of a regular polygon with perimeter [tex]\( P \)[/tex] and apothem length [tex]\( a \)[/tex] is:
[tex]\[ A = \frac{1}{2} (P \times a) \][/tex]

Thus, the proper choice among the options is:
[tex]\[ \text{Option A: } A = \frac{1}{2} (P \times a) \][/tex]