Question

Point $O$ is the center of both circles in the figure above. If the circumference of the large circle is $36$ and the radius of the small circle is half of the radius of the large circle, what is the length of the smaller arc of the smaller circle?

• A. $10$
• B. $8$
• C. $6$
• D. $4$

Answer 1

The correct option is D $4$

Let ${}^{\prime }{r}^{\prime }$ be the radius of the smaller circle

Let ${}^{\prime }{R}^{\prime }$ be the radius of the large circle

Given, circumference of the large circle $=$ $36$

$\Rightarrow 2\pi R$ $=$ $36$

$\Rightarrow R$ $=$ $\frac{36}{2\pi }$

$\Rightarrow R$ $=$ $\frac{18}{\pi }$

As radius of smaller circle $r$ is half of the radius of the large circle $R$,

$\Rightarrow r$ $=$ $\frac{R}{2}$

$\Rightarrow r$ $=$ $\frac{\frac{18}{\pi }}{2}$

$\Rightarrow r$ $=$ $\frac{18}{2\pi }$

$\Rightarrow r$ $=$ $\frac{9}{\pi }$

We know that, if ${}^{\prime }{x}^{\prime }$ is the angle subtended by an arc at the center of the circle and $r$ be the radius of the circle.

Length of the arc $=$ $\frac{x}{360}$ $\times$ $2\pi r$

Here $x$ $=$ $80$ and $r$ $=$ $\frac{9}{\pi }$,

Length of the darkened arc $=$ $\frac{x}{360}$ $\times$ $2\pi r$

$=$ $\frac{80}{360}$ $\times$ $2\times \pi$ $\times$ $\frac{9}{\pi }$

$=$ $4$

Therefore, Length of the darkened arc is ${}^{\prime }{4}^{\prime }$.