Question

A circle of radius $4\,cm$is cut out through a square sheet of $12\,cm$. Find the area of the remaining sheet. Also, find the cost of a sheet at $Rs\,8$ per $c{m^2}$.

Answer 1

Hint: Here in this question we want to find the area of a square sheet from which a circular portion has been cut out whose radius is $4\,cm$. So, to calculate the area of the remaining sheet, we have to first calculate the area of the whole sheet and then calculate the area of the circular portion that has been cut out from the sheet. Then, we subtract the area of the circular portion from the square sheet so as to find the area of the remaining sheet. We know the value of $\pi $ and the value of radius of the circle and side of the square are given to us in the question itself. We substitute the known values and determine the areas of both the figures using the appropriate formulae.

Complete step-by-step solution:
Firstly, we have to find the area of the circular portion and square sheet.
So, the circle is a two dimensional figure and we have to determine the area, where area is the region or space occupied by the circular field. To determine the area of a circle we have standard formula $A = \pi {r^2}$ where r represents the radius. The radius of a circle is the line segment which joins the centre of the circle to the any point on the circle or to the circumference. The unit for the area is square units. In the given question, we are given the length of the radius in centimetres. So, we get the area of the circle using the formula in the unit $c{m^2}$.
To find the area of a circle, we use formula$A = \pi {r^2}$. The radius of circle is given as$4\,cm$.
By substituting the value of radius of the circle, we get,
$A = \pi {r^2}$
$ \Rightarrow A = \pi {\left( 4 \right)^2}\,c{m^2}$
$ \Rightarrow A = 16\pi \,c{m^2}$
We can substitute the value of $\pi $ to find the area and we can simplify further.
Substituting the value of $\pi $, we have,
$ \Rightarrow A = 16\left( {3.14} \right)c{m^2}$
Further simplifying the calculations, we have,
$ \Rightarrow A = 50.24\,c{m^2}$
Hence the area of a circle whose radius is $4$ centimetres is $50.24\,c{m^2}$.
Side of square$ = 12\,cm$
Now, area of square $ = {\left( {Side} \right)^2} = 144c{m^2}$
Now, the area of the remaining sheet$ = \left( {Area\,of\,square} \right)\, - \,\left( {Area\,of\,circle} \right)$
$ \Rightarrow Area\,of\,remaining\,sheet = 144\,c{m^2} - 50.24\,c{m^2} = 93.76\,c{m^2}$
So, the area of remaining sheet is $93.76\,c{m^2}$.
Rate of the sheet$ = Rs\,8$per $c{m^2}$
Also, the cost of sheet$ = Rs\,8/c{m^2} \times 93.76c{m^2}$
$ = Rs\,750.08$

So, the cost of remaining sheet is $Rs\,750.08$

Note: A circle is a closed two dimensional figure. Generally the area is the region occupied by the thing. The area of a circle is defined as the region occupied by the circular region. It can be determined by using formula $A = \pi {r^2}$ where r is the radius of the circle. The radius is denoted by r or R.