Question

In the given figure, the side of the square is $28$ cm and the radius of each circle is half of the length of the side of the square where $O$ and $O^{\prime }$ are centers of circles. Find the area of the colored portion.

Answer 1

Hint: The given geometric figure deals with the shapes called circle and square. We will use the area formulas of these shapes to find the area of the required portion. More precisely we will add the area of the circles and square and then subtract the part that occurred twice.
Formula used:
1) The area of a square of side $a$ is $a^2$
2) The area of a circle of radius $r$ is $\pi r^2$ , where $\pi =3.141$

Complete step-by-step answer:
The given figure has a square of side $28$ cm.
Let $2r=28$ --(1)
Let the radius of the circle with Centre $O$ and $O^{\prime }$ be $r_1$ and $r_2$ respectively.
According to the question the radius of the two circles is half of the side of the square.
So, $r_1=r_2={\displaystyle \frac{1}{2}}\times 2r=r$

Let $A_s$ represent area of square, $A_O$ represent area of the circle with Centre $O$ and $${A_O}^1$$ represent area of the circle with Centre $O^1$
Now from the formulas of area of these geometric shapes we have:
 $A_s=(2r{)}^2$
 $A_O=\pi r^2$
${A_O}^1=\pi r^2$
Adding all the above we get total area $A$ as:
 $A=(2r{)}^2+\pi r^2+\pi r^2$
The area we have calculated adds the area common between the two circles and square.
From the figure we can see that the extra added area is ${\displaystyle \frac{1}{4}}$ th the area of the circles.
So we have to subtract it from the generated area in order to find the required area.
Now, the required area becomes
 $A=(2r{)}^2+\pi r^2+\pi r^2-{\displaystyle \frac{\pi r^2}{4}}-{\displaystyle \frac{\pi r^2}{4}}$
 $\Rightarrow A=4r^2+2\pi r^2-2\times {\displaystyle \frac{\pi r^2}{4}}$
 $\Rightarrow A=(4+2\pi )r^2-{\displaystyle \frac{\pi r^2}{2}}$
 $\Rightarrow A=(4+2\pi -{\displaystyle \frac{\pi }{2}})r^2$
 $\Rightarrow A=({\displaystyle \frac{8+4\pi -\pi }{2}})r^2$
 $\therefore A={\displaystyle \frac{8+3\pi }{2}}\times r^2$
From (1) we have $2r=28\Rightarrow r=14$ so the above equation becomes:
 $A={\displaystyle \frac{8+3\times 3.14}{2}}\times (14{)}^2$
 $A={\displaystyle \frac{8+9.42}{2}}\times 196$
 $A=17.42\times 98=1707.16$ $c{m}^2$
So, the correct answer is “ 1707.16 $c{m}^2$”.

Note: The area common between the circle two circles and the square will give an area greater than the area of the colored region. To avoid this we have to subtract the repeated area while calculating the required shaded area. Don’t forget to put the unit of the area after calculating it, in our case it is $c{m}^2$