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]
