Circle Square Area Comparison

This analysis compares the areas of a circle and a square when their perimeters are equal, focusing on the relationship stated in the provided options.

Perimeter Setup

Let $P$ represent the common perimeter of both the circle and the square.

Square Area Calculation

For a square with perimeter $P$:

• The side length ($s$) is calculated as $s = \frac{P}{4}$.
• The area ($A_s$) is $A_s = s^2 = \left(\frac{P}{4}\right)^2 = \frac{P^2}{16}$.

Circle Area Calculation

For a circle with circumference $P$:

• The radius ($r$) is found using the formula $P = 2\pi r$, which gives $r = \frac{P}{2\pi}$.
• The area ($A_c$) is $A_c = \pi r^2 = \pi \left(\frac{P}{2\pi}\right)^2 = \pi \left(\frac{P^2}{4\pi^2}\right) = \frac{P^2}{4\pi}$.

Area Relationship Statement

The question requires determining the relationship between the area of the circle ($A_c$) and the area of the square ($A_s$) given they have the same perimeter. Based on the provided correct answer (Option D):

The area of the circle is two times the area of the square.

This relationship is expressed mathematically as $A_c = 2 \times A_s$.