Question

What is the minimum radius $(>1)$ of a circle whose circumference is an integer?

• A. $2$
• B. $\frac{4}{\pi }$
• C. $\frac{6}{\pi }$
• D. $\frac{3}{\pi }$

Answer 1

Answer: (B)

$\frac{4}{\pi }$

Circumference of circle $=$ $2\pi r$

For integer, value of $r$ must be an integer multiple of $\frac{1}{\pi }$.
Let $r=\frac{x}{\pi }$, where $x$ is a positive integer.
But $r>1$

$\therefore \frac{x}{\pi }>1$
$\Rightarrow x>\pi \Rightarrow x>3.14$
For minimum $r$, $x$ must be minimum
Since least integer greater than $\pi$ is $4$.

$\therefore x=4$
Minimum value of $r=\frac{4}{\pi }$.