Question

A wire of length 36 cm is bent in the form of a semicircle. What is the radius of the semicircle?

Answer 1

Hint: In this question we will see some facts about semicircle. A semicircle is a half-circle. You might think that the perimeter of a semicircle is half of the perimeter of a circle. But it not true. The perimeter of a semicircle is half of the original circle's circumference, plus the diameter. Since the semicircle includes a straight line that is diameter, we have to include the diameter while calculating its perimeter. So, the perimeter of a semicircle is $\pi r + d$, where $r$ = radius and $d$ = diameter of a semicircle.

Complete step by step solution:
Given, The length of a wire = 36 cm
The length of a wire which bent in the form of a semicircle is equal to the perimeter of a semicircle.

Perimeter of semicircle (Or half circle) is $\dfrac{1}{2}$ × (circumference of circle) + length of diameter
Now, according to the question,
Length of wire = Perimeter of semicircle
Perimeter of a semicircle =$\pi r + d$
$ = \pi r + 2r = 36$
$ = r(\pi + 2) = 36$(Taking \[\pi \] common)
$ = \left( {\pi + 2} \right) = \dfrac{{36}}{r}$
$ = \dfrac{{(\pi + 2)}}{{36}} = \dfrac{1}{r}$
\[ = \dfrac{{\dfrac{{22}}{7} + 2}}{{36}} = \dfrac{1}{r}\](Putting the value of \[\pi \])
$\Rightarrow r = 7 cm$ (After solving the above equation we get the value of $r$)

$\therefore$ The radius of the semicircle is 7cm.

Note:
In these types of questions, while calculating the perimeter of a semicircle we need to keep a few things in our mind precisely. If you think that half of the perimeter of a circle is the perimeter of a semicircle then you are wrong. If you look at the semicircle then you will notice that its perimeter must include the round portion and the diameter of a semicircle.