To determine the formula for finding the area of a regular polygon with perimeter [tex]\(P\)[/tex] and apothem length [tex]\(a\)[/tex], let's break down the steps and understand the geometric principles involved.
1. Understanding the Variables:
- [tex]\(P\)[/tex] is the perimeter of the regular polygon.
- [tex]\(a\)[/tex] is the apothem, which is the distance from the center of the polygon to the midpoint of one of its sides.
2. Area of a Regular Polygon:
- A regular polygon can be divided into [tex]\(n\)[/tex] isosceles triangles, where [tex]\(n\)[/tex] is the number of sides.
- Each triangle has a base equal to the length of a side of the polygon and a height equal to the apothem [tex]\(a\)[/tex].
3. Calculating the Area of One Triangle:
- The area [tex]\(A_{\triangle}\)[/tex] of one triangle is given by [tex]\(\frac{1}{2} \times \text{base} \times \text{height}\)[/tex].
4. Total Perimeter as Sum of Bases:
- The perimeter [tex]\(P\)[/tex] of the polygon is the sum of the bases of all the triangles.
5. Total Area of the Polygon:
- Since the polygon consists of [tex]\(n\)[/tex] triangles, the total area [tex]\(A\)[/tex] of the polygon is the sum of the areas of all these triangles.
- Hence, the total area [tex]\(A\)[/tex] can be written as:
[tex]\[
A = n \times A_{\triangle} = n \times \left(\frac{1}{2} \times \text{base} \times a\right)
\][/tex]
- Since the sum of the bases of all triangles is the perimeter [tex]\(P\)[/tex]:
[tex]\[
A = \frac{1}{2} \times P \times a
\][/tex]
Therefore, the correct formula for finding the area of a regular polygon with perimeter [tex]\(P\)[/tex] and apothem length [tex]\(a\)[/tex] is:
[tex]\[ A = \frac{1}{2} P a \][/tex]
This matches with option C:
[tex]\[ \text{C. } A = \frac{1}{2} (P a) \][/tex]
Thus, the correct answer is C.
