Question

If the perimeter of a semi-circle is 36 cm. what will be its diameter?
A. 88 cm
B. 22 cm
C. 28 cm
D. 14 cm

Answer 1

Hint: As we all know that the perimeter of a circle is \[2\pi R\] where ‘R’ is the radius of the circle. Similarly, the perimeter of a semi-circle is half the perimeter of the circle with the addition of a length of the diameter which twice as the radius. With consideration of all these points, we’ll form an equation in ‘R’ to find its value and further to get the value of diameter.

Complete step by step answer:

Given data: Perimeter of semi-circle=36 cm.

Using Diameter(D) of semi-circle= 2[radius(R) of semi-circle]
The perimeter of the semi-circle${\text{ = $\pi$ R + D}}$
$
   \Rightarrow {\text{$\pi$ R + D = $\pi$ R + 2R}} \\
   \Rightarrow {\text{R(2 + $\pi$ )}} \\
 $
It is given that
Perimeter of semi-circle=36 cm
$
   \Rightarrow {\text{R(2 + $\pi$ ) = 36}} \\
  \therefore {\text{R = }}\dfrac{{{\text{36}}}}{{{\text{2 + $\pi$ }}}} \\
 $
Putting the value of ${\text{$\pi$ = }}\dfrac{{{\text{22}}}}{{\text{7}}}$ , we get
$ \Rightarrow {\text{R = }}\dfrac{{{\text{36}}}}{{{\text{2 + }}\dfrac{{{\text{22}}}}{{\text{7}}}}}$
On simplification we get,
$ \Rightarrow {\text{R = }}\dfrac{{{\text{36(7)}}}}{{{\text{14 + 22}}}}$
$
   \Rightarrow {\text{R = 36 $\times$ }}\dfrac{{\text{7}}}{{{\text{36}}}} \\
  \therefore {\text{R = 7}} \\
 $
Since, Diameter(D) of semi-circle= 2[radius(R) of semi-circle]
$
  {\text{D = 2R}} \\
  {\text{ = 2(7)}} \\
  {\text{ = 14cm}} \\
 $
Therefore, (D) 14 cm is the correct answer.

Note: We can approach the answer using the formula for the perimeter as
The perimeter of the semi-circle${\text{ = $\pi$ }}\dfrac{D}{2}{\text{ + D}}$
${\text{$\pi$ }}\dfrac{{\text{D}}}{{\text{2}}}{\text{ + D = D(1 + }}\dfrac{{\text{$\pi$ }}}{{\text{2}}}{\text{)}}$
It is given that
Perimeter of semi-circle=36 cm
$
  {\text{D(1 + }}\dfrac{{\text{$\pi$ }}}{{\text{2}}}{\text{) = 36}} \\
   \Rightarrow {\text{D = }}\dfrac{{{\text{36}}}}{{{\text{1 + }}\dfrac{{\text{$\pi$ }}}{{\text{2}}}}} \\
 $
Putting the value of ${\text{$\pi$ = }}\dfrac{{{\text{22}}}}{{\text{7}}}$ , we get
$
  {\text{D = }}\dfrac{{{\text{36}}}}{{{\text{1 + }}\dfrac{{{\text{11}}}}{{\text{7}}}}} \\
   \Rightarrow {\text{D = 36 $\times$ }}\dfrac{{\text{7}}}{{{\text{18}}}} \\
   \Rightarrow {\text{D = 2 $\times$ 7}} \\
  \therefore {\text{D = 14}} \\
 $
Therefore, Diameter of semi-circle= 14 cm