Question : If the perimeter of circle A is equal to the perimeter of semicircle B, what is the ratio of their areas?
Option 1: $\left (x+ 2\right)^{2} : 2\pi ^{2}$
Option 2: $2\pi ^{2} : \left ( x+2 \right)^{2}$
Option 3: $\left (\pi +2 \right )^{2} : 4\pi ^{2}$
Option 4: $4\pi ^{2}:\left ( \pi +2 \right)^{2}$
Correct Answer: $\left (\pi +2 \right )^{2} : 4\pi ^{2}$
Solution :
Let the radius of circle A be r
1
units and the radius of semicircle B is r
2
units.
According to the question,
2$\pi$r
1
= $\pi$r
2
+ 2r
2
⇒ 2$\pi$r
1
= r
2
($\pi$+2)
⇒ $\frac{r_{1}}{r_{2}}=\frac{\pi+2}{2\pi}$
$\therefore$ Ratio of their areas = $\frac{\pi(r_{1})^{2}}{\pi(r_{2})^{2}} = \frac{(\pi+2)^{2}}{(2\pi)^{2}}=(\pi+2)^{2} : 4\pi^{2}$
Hence, the correct answer is $(\pi+2)^{2} : 4\pi^{2}$.
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