Question

Calculate the area of the designed region in figure common between the two quadrants of circle of radius 8 cm each.

Answer 1

Hint: In this problem, we have to find the area of the designed region in the given figure common between the two quadrants of a circle of radius 8 cm each. Here we can find the area of the first quadrant and the area of the second quadrant. We can add them and subtract the result with the area of the square to get the area of the designed region. We can use the formula for area of quadrant, \[\dfrac{\theta }{{{360}^{\circ }}}\times \pi {{r}^{2}}\] as the given radius is 8 cm. We can find the area of the square with \[{{a}^{2}}\] formula.

Complete step by step solution:
Here we have to find the area of the designed region in the given figure common between the two quadrants of a circle of radius 8 cm each and the angle is \[{{90}^{\circ }}\].

We can see from the diagram that,
Area of the designed region = Area of the first quadrant + Area of the second quadrant – Area of square.
We can now find the area of the first quadrant and the area of the second quadrant.
Area of first quadrant = \[\dfrac{\theta }{{{360}^{\circ }}}\times \pi {{r}^{2}}=\dfrac{{{90}^{\circ }}}{{{360}^{\circ }}}\times \pi {{\left( 8 \right)}^{2}}\]
We can now solve the above step, we get
Area of first quadrant = \[\dfrac{{{90}^{\circ }}}{{{360}^{\circ }}}\times \pi {{\left( 8 \right)}^{2}}=\dfrac{1}{4}\times \dfrac{22}{7}\times 64=\dfrac{352}{7}c{{m}^{2}}\].
Since, the radius and angles are equal, we have
Area of the second quadrant = Area of first quadrant = \[\dfrac{352}{7}c{{m}^{2}}\].
We can now find the area of square,
 Area of square = \[{{a}^{2}}=8\times 8=64\].
We can now find the area of the designated region.
 Area of the designed region = \[\dfrac{352}{7}+\dfrac{352}{7}-64\].
We can now simplify the above step, we get
 Area of the designed region = \[\dfrac{704-448}{7}=\dfrac{256}{7}c{{m}^{2}}\].
Therefore, the area of the designed region is \[\dfrac{256}{7}c{{m}^{2}}\].

Note: We should remember some formula like area of quadrant is \[\dfrac{\theta }{{{360}^{\circ }}}\times \pi {{r}^{2}}\] and area of square is square of the length of the side, \[{{a}^{2}}\]. We should also concentrate on the given data and the diagram and understand what is being asked to find the result for the given problem.