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# 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′ are centers of the circles. Find the area of the shaded region.

Last updated date: 08th Aug 2024
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Hint:
Here we have to find the area of the shaded region. For that, we will first find the area of the square. Then we will find the area of the two circles whose radius is equal to half of the length of the square. We will find the overlapping area, which is the quadrant of a circle and the area of the quadrant is equal to the one-fourth area of a circle. For the resulting shaded area, we will add the area of square and area of two circles and then we will subtract the overlapping area from the sum. The resulting area will be the required shaded area.

Complete step by step solution:
We have to find the area of the shaded region here.
We will first find the area of a square of side 28 cm.
We know the area of the square $= {\left( {side} \right)^2}$.
Substituting the value of side in the area of square, we get
$\Rightarrow$ Area of square$= {\left( {28cm} \right)^2} = 784c{m^2}$
We know, the radius of a given circle is half of the given side of a square i.e.
$\Rightarrow r = \dfrac{{28}}{2} = 14cm$
Now, we will calculate the value of the area of the circle.
We know;
Area of circle $= \pi {r^2}$
Substituting the value radius here, we get
$\Rightarrow$ Area of circle $= \pi {\left( {14} \right)^2}$
On multiplying the terms, we get
$\Rightarrow$ Area of circle $= 616c{m^2}$
Thus, area of both the circles $= 2 \times 616 = 1232c{m^2}$
Now, we will calculate the area of the two quadrants, which is equal to one-fourth area of a circle.
Thus, the area of two quadrants $= 2 \times \dfrac{1}{4} \times 616 = 308c{m^2}$
Thus,
Area of the shaded region $=$ area of two circles $+$ area of square $-$ area of two quadrants we will
Substituting all the values of different areas in the above equation, we get
$\Rightarrow$ Area of the shaded region $= 1232 + 784-308 = 1708{\rm{c}}{{\rm{m}}^2}$

Hence, the area of the shaded region is $1708c{m^2}$.

Note:
Here we have subtracted the area two quadrants because on the addition of area of square and area of two circles we would have got the area of these two quadrants two times. The overlapped area is quadrant because one of the vertices of the square is at the center of the circle, which is forming $90^\circ$ at the center.