Question

If the circumference of a circle of radius $r$ is equal to the perimeter of a semicircle of radius $R$, then $R:r$ equals

• A. $2:1$
• B. $\pi :(\pi +2)$
• C. $2\pi :(\pi +2)$
• D. $2\pi :(\pi +1)$

Answer 1

The correct option is C $2\pi :(\pi +2)$

Circumference of circle of radius $r=2\pi r$
Perimeter of semicircle of radius, $R=\frac{2\pi R}{2}+2R$
$=R(\pi +2)$

Now, Circumference of circle of radius $r=$ Perimeter of semicircle of radius $R$,
$\Rightarrow R(\pi +2)=2\pi r$
$\Rightarrow \frac{R}{r}=\frac{2\pi }{\pi +2}$
$\Rightarrow R:r=2\pi :(\pi +2)$
Hence, the correct answer is option c.