Question

The shape of the garden is rectangular in the middle and semi-circular at the ends as shown in the diagrams. Find the area and the perimeter of this garden. [Length of rectangle is $20 - \left( {3.5 + 3.5} \right)$ metres]

Answer 1

Hint: First of all, find the area of the rectangle using the formula $l \times b$, where $l$ is the length and $b$ is the breadth of the rectangle. Then, find the area of semicircles and add it with the area of the rectangle to find the total area of the garden. Next, find the length of the boundary of the garden as the perimeter of the garden.

Complete step by step solution: We are given the shape of the garden.
The garden has a rectangular shape in the middle and it is semi-circular at the ends.
We have to find the area and the perimeter of the garden.
We find the area and the perimeter of the garden.
First of all, we will find the area of the garden.
The area of the garden will be equal to the area of the rectangle and the area of the two semi-circles at the end of the garden.
We will have to calculate the length and breadth of the rectangular portion of the garden.
As, the diameter of the semicircles, is 7m the radius will be 3.5m because the radius is half the diameter.
Then, the length of the rectangular portion is
$
  20 - \left( {3.5 + 3.5} \right) = 20 - 7 \\
   = 13m \\
$

And the breadth of the rectangle is 7m.
Now, the area of a rectangle is the product of its length and breadth.
Hence, the area of the rectangular portion is $13 \times 7 = 91{m^2}$
Now, calculate the area of the semicircle with radius 3.5m
The area of the semicircle is $\dfrac{{\pi {r^2}}}{2}$
On substituting the value of $\pi = \dfrac{{22}}{7}$ and $r = 3.5$ in the above formula, we get,
$\dfrac{{\dfrac{{22}}{7}{{\left( {3.5} \right)}^2}}}{2} = \dfrac{{22 \times 3.5 \times 3.5}}{{7 \times 2}}$
On solving the above expression, we get
$\dfrac{{22 \times 3.5 \times 3.5}}{{7 \times 2}} = 19.25{m^2}$
The area of the semicircle will also be same.
Thus, the area of garden is the summation of areas of rectangular portion and area of semicircles.
Hence, the area of the garden is $91 + 19.25 + 19.25 = 129.5{m^2}$
Now, let us find the perimeter of the garden.
First of all, find the perimeter of the semicircles.
Perimeter of semicircle is given by $\pi r$
On substituting the value of $\pi = \dfrac{{22}}{7}$ and $r = 3.5$ in the above formula, we get,
$\dfrac{{22}}{7}\left( {3.5} \right) = 11m$
Perimeter of the other semicircle will also be same.
Hence, the perimeter of other semicircle is also 11m.
We know that the perimeter of any shape is defined as the length of its boundary.
Then, the boundary of the garden includes the length of the rectangle and not the breadth of the rectangle.
The total perimeter of the garden is $11 + 11 + 13 + 13 = 48m$

Hence, the area of the garden is $129.5{m^2}$ and the perimeter of the garden is 48m.

Note: In the question of mensuration we first need to check which formula we can use to solve the given question. In these kinds of questions, formula plays a crucial role in solving the problem. For example, in this question, they can give any figure and can ask us to find the area. We just need to divide the total area into subparts for which we can calculate area by using formulas of our knowledge.