Certainly! The problem asks us to find the formula for the area [tex]\( A \)[/tex] of a regular polygon when given its perimeter [tex]\( P \)[/tex] and apothem length [tex]\( a \)[/tex].
To derive this formula, let's break it down step-by-step:
1. Understanding Terms:
- Regular Polygon: A polygon with all sides and all angles equal.
- Perimeter ([tex]\( P \)[/tex]): The total length around the polygon.
- Apothem ([tex]\( a \)[/tex]): The distance from the center of the polygon to the midpoint of one of its sides.
2. Area Formula of a Regular Polygon:
- For a regular polygon, the area [tex]\( A \)[/tex] can be calculated using the apothem ([tex]\( a \)[/tex]) and the perimeter ([tex]\( P \)[/tex]).
- The formula is derived by considering the polygon as being composed of identical isosceles triangles, each with a base as one side of the polygon and height as the apothem.
3. Detailed Formula Derivation:
- Each of the [tex]\( n \)[/tex] triangles (where [tex]\( n \)[/tex] is the number of sides in the polygon) has a base [tex]\( \frac{P}{n} \)[/tex] and height [tex]\( a \)[/tex].
- The area of one such triangle is [tex]\( \frac{1}{2} \times \frac{P}{n} \times a \)[/tex].
- Since there are [tex]\( n \)[/tex] such triangles, the total area [tex]\( A \)[/tex] is:
[tex]\[
A = n \times \left( \frac{1}{2} \times \frac{P}{n} \times a \right)
\][/tex]
- Simplifying this gives:
[tex]\[
A = \frac{1}{2} P a
\][/tex]
Therefore, the correct formula for finding the area of a regular polygon with perimeter [tex]\( P \)[/tex] and apothem [tex]\( a \)[/tex] is:
[tex]\[
A = \frac{1}{2} P a
\][/tex]
Hence, the correct option is:
C. [tex]\( A = \frac{1}{2} (P a) \)[/tex]
