Question

The figure above consists of a square and 2 semicircle, with dimensions as shown. Calculate its outside perimeter (in centimeters)

$\begin{align}
  & \text{A}\text{. }8+8\pi \\
 & \text{B}\text{. }16+8\pi \\
 & \text{C}\text{. }16+16\pi \\
 & \text{D}\text{. }32+8\pi \\
 & \text{E}\text{. }32+16\pi \\
\end{align}$

Answer 1

Hint: To find the perimeter of the given figure, first we will find the outer surface of the semicircle which is also known as the circumference of a half circle. Then we will find the perimeter of the figure by adding all the sides of the figure.

Complete step by step solution:

From the figure, we get to know that –
It consists of a square and 2 semicircles and the side of a square is equal to the diameter of the semicircle. i.e. $s=d=8$ …….. (1)
We know that –
$2\text{ Semicircles = Circle}$
Dividing both sides by 2, we get –
$\text{Semicircle=}\dfrac{1}{2}\text{Circle}$ …………………. (2)
We know that the perimeter of a circle is also known as circumference of a circle.
So the circumference of a $\dfrac{1}{2}$ circle $=\dfrac{1}{2}2\pi r$
Where, $r=\dfrac{d}{2}$ and $d=8$ (from equation 1)
So the circumference of a $\dfrac{1}{2}$ circle $=\dfrac{1}{2}\times 2\times \pi \times \dfrac{8}{2}$
By cancelling the common factor from numerator and denominator, we get –
$=4\pi $
Therefore, from equation (2) –
$\text{circumference of }\dfrac{1}{2}\text{ circle = circumference of semicircle =4}\pi $ …………………… (3)
Now, we will find the outside perimeter of the given figure.
The perimeter is the sum of all sides of the figure,
Let us consider, ‘s’ as the sides of squares and ‘c’ is the circumference of the semicircle.

The perimeter of the given figure is $s+c+s+c$ .
By substituting the values from equation 1 and 3, we get that –
$\begin{align}
  & 2s+2c=2\left( 8 \right)+2\left( 4\pi \right) \\
 & =16+8\pi \\
\end{align}$
Therefore, the perimeter of the given figure is $16+8\pi $ .
Hence, the correct option is B.

Note: To solve this problem students must know that the length of a straight sided shapes outline is called perimeter, and the length of a circle’s outline is called its circumference. Here in this question there are 2 semi circles which is equal to a circle of same dimension
For example:

From the figure we can clearly see that the 2 semicircles of the same radius are equal to the circle of the same radius.
So, to find the perimeter of a circle we must need to find the circumference of semicircles.