Question

what is the formula for finding the area of a regular polygon with perimeter ( p ) and apothem length ( a )?
a. ( a = \frac{1}{2}(pa) )
b. ( a = pa )
c. ( a = pa )
d. ( a = \frac{1}{2}(pa) )

Explanation:

Step1: Recall regular polygon area formula

A regular polygon can be divided into congruent isosceles triangles, each with base equal to a side length $s$ and height equal to the apothem $a$. The area of one triangle is $\frac{1}{2}sa$.

Step2: Relate to perimeter

The perimeter $P$ of the polygon is $n \times s$, where $n$ is the number of sides. The total area is $n \times \frac{1}{2}sa = \frac{1}{2}(ns)a$. Substitute $ns = P$:
$A = \frac{1}{2}(Pa)$

Step3: Match to options

This matches option A, while other options incorrectly relate variables or omit the $\frac{1}{2}$ factor.

Answer:

A. $A = \frac{1}{2}(Pa)$