Question

Given that the circle is inscribed in the given square, would the ratio of the cicumference of the circle to the perimeter of the square or the ratio of the area of the circle to the area of the square change if the length of the square were to be changed?

• A. Yes
• B. No
• C. Cannot Establish

Answer 1

The correct option is B No

To see if the ratios would change, we must first establish a relationship between the side of the square to the radius of the circle. Since the circle is inscribed in the square, we can say that $r=\frac{s}{2}$ where $r$ is the radius of the circle and $s$ is the side of the square.

We can also say that from here, $s=2r$

Ratio 1: Comparing perimeters
$r_{perimeter}=\frac{2\pi r}{4s}$
Substituting $s=2r$, we get:
$=\frac{2\pi r}{8r}$
$=\frac{\pi }{4}$
which is a constant independent of the length of side of given square

Ratio 1: Comparing areas
$r_{area}=\frac{\pi r^2}{s^2}$
Substituting $s=2r$, we get:
$=\frac{\pi r^2}{(2r{)}^2}$
$=\frac{\pi }{4}$

which is a constant independent of the length of side of given square

Hence the ratios do not change!