Question

A circular wire of radius $(2+\pi )$ cm is cut and bent into a semicircle. Find the radius of the semicircle.

• A. $2\pi \text{cm}$
• B. $2(2+\pi )\text{cm}$
• C. $4\text{cm}$
• D. $4\pi \text{cm}$

Answer 1

The correct option is A $2\pi \text{cm}$

Given, the radius of the circle $=(2+\pi )\text{cm}$
Perimeter of the circle
$=2\pi \times \text{radius}=2\pi (2+\pi )$

Circle is bent into a semicircle.
Let the radius of the semicircle be $r_1$.
Perimeter of the semicircle $=\text{diameter + half of the circumference of a circle}$
$=2r_1+\frac{2\pi r_1}{2}$
$=2r_1+\pi r_1$
Taking $r_1$ common
$=r_1(2+\pi )$


As length of the wire is constant, perimeter of both shapes would be same.
$\Rightarrow \text{Perimeter of the circle = Perimeter of the semicircle}$
$\Rightarrow 2\pi (2+\pi )=r_1(2+\pi )$
Cancel out $(2+\pi )$ from both sides
$\Rightarrow 2\pi =r_1$
$\Rightarrow r_1=2\pi$

Hence, the radius of the semicircle is $2\pi \text{cm}.$