## Task 5 - 230835

a) For a given circle $K$, consider the square $Q$ that has the same perimeter as $K$.

Is the area of $Q$ larger, equal, or smaller than the area of $K$? What percentage of the area of $K$ is the area of $Q$?
b) For a given circle $k$, consider the square $q$ that has the same area as $k$.

Is the perimeter of $q$ larger, equal, or smaller than the perimeter of $k$? What percentage of the perimeter of $k$ is the perimeter of $q$?

The approximate value of $\pi$ accurate to four decimal places, $\pi \approx 3.1416$, can be used. The required percentages should be given to one decimal place after the decimal point.
a) If $K$ has the radius $r$ and $Q$ the side length $a$, then by assumption $4 a=2 \pi r$, so $a=\frac{\pi}{2} r$. The area of $K$ is $J(K)=\pi r^{2}$.

The area of $Q$ is $J(Q)=a^{2}=\frac{\pi}{4} r^{2}$, which is $\frac{\pi}{4}$ times, or $25 \pi \% \approx 78.5 \%$ of $J(K)$.

The area of $Q$ is thus smaller than that of $K$.

b) If $k$ has the radius $\rho$ and $q$ the side length $s$, then by assumption $s^{2}=\pi \rho^{2}$, so $s=\sqrt{\pi} \rho$.

The circumference of $k$ is $2 \pi \rho$. The circumference of $q$ is $4 s=4 \sqrt{\pi} \rho$, which is $\frac{2}{\sqrt{\pi}}$ times, or $\approx 112.8 \%$ of the circumference of $k$.

The circumference of $q$ is thus larger than that of $k$.