Question

The shape of a garden is rectangular in the middle and semi-circular at the ends as shown in the diagram. Find the area and the perimeter of this garden. [length of the rectangle is 20-(3.5+3.5)metres].

A) $129.5{\text{ }}{m^2};{\text{ }}48{\text{ }}m$
B) $179.5{\text{ }}{m^2};{\text{ }}48{\text{ }}m$
C) $129.5{\text{ c}}{m^2};{\text{ }}48{\text{ }}m$
D) $129.5{\text{ }}{m^2};{\text{ 8}}8{\text{ c}}m$

Answer 1

Hint:
Observe the figure and determine the figures put together to form that image. Then apply the formula of perimeter and area to find their individual perimeter and areas and then finally add their areas and perimeter to find the perimeter and area of the figure.

Complete step by step solution:
Let us consider the given figure and observe that it has 2 semi-circles attached both the ends and in between there is a rectangle.
Let us first find the perimeter of the figure. For that, we find the perimeter of the two semi-circles and the perimeter of the rectangle and then add them.
From the figure, it is seen that the diameter of the semi-circle is 7 metres, and it is known that the radius of the semi-circle is half of the diameter.
$
   \Rightarrow r = \dfrac{7}{2}{\text{ metres}} \\
   \Rightarrow r{\text{ = 3.5 metres}} \\
 $
Find the combined circumference ( as two semi-circles of the same radius form a complete circle).
$
   \Rightarrow C = 2\pi r \\
   \Rightarrow C = 2 \times \dfrac{{22}}{7} \times 3.5 \\
   \Rightarrow C = 22{\text{ metres}} \\
 $
Also, it is given length of the rectangle is 20-(3.5+3.5) metres. Simplify it to find its accurate value.
$
  20 - \left( {3.5 + 3.5} \right) = 20 - 7 \\
   = 13{\text{ metres}} \\
 $
Now, we will do the twice of the length of the rectangular portion,
Thus, we get,
\[ \Rightarrow 2 \times 13 = 26\]
Now, add the combined circumference and the length of the rectangle to find the perimeter of the figure.
$
   \Rightarrow P = 22{\text{ metres}} + 26{\text{ metres}} \\
   \Rightarrow P = 48{\text{ metres}} \\
 $
Next, let us find the area of the figure. For that, we find the area of the two semi-circles and the area of the rectangle and then add them.

Find the combined area ( as two semi-circles of the same radius form a complete circle).
$
   \Rightarrow A = \pi {r^2} \\
   \Rightarrow A = \dfrac{{22}}{7} \times 3.5 \times 3.5 \\
   \Rightarrow A = 38.5{\text{ metre}}{{\text{s}}^2} \\
 $
From the figure, we can see that the length of the rectangle is 13 metres and the width is 7 metres. Thus, find the area of the rectangle,
$
   \Rightarrow A = l \times w \\
   \Rightarrow A = 13 \times 7 \\
   \Rightarrow A = 91{\text{ metre}}{{\text{s}}^2} \\
 $
Now, add the combined area and the area of the rectangle to find the area of the figure.
$
   \Rightarrow A = 91{\text{ metre}}{{\text{s}}^2} + 38.5{\text{ metre}}{{\text{s}}^2} \\
   = 129.5{\text{ metre}}{{\text{s}}^2} \\
 $

Hence, the given figure has an area of $129.5{\text{ }}{{\text{m}}^2}$ and the perimeter of $48{\text{ m}}$.
Hence, option (A) is the correct option.

Note:
Remember to subtract the region or length which is already covered in the dimension of the other figures. In other words, you need to be sure that there is no overlapping of the area that you have calculated.