Ex.12.3 Q16 Areas Related to Circles Solution - NCERT Maths Class 10

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Question

Calculate the area of the designed region in Figure common between the two quadrants of circles of radius \(\text{8 cm}\) each.

Text Solution

What is  known?

Designed region is the common area between the \(2\) quadrants of circles of radius \(\text{8 cm.}\)

What is unknown?

Area of the designed region.

Reasoning:

In a circle with radius r and angle at the center with degree measure \(\theta\);

(i) Area of the sector \(\begin{align} = \frac{{\rm{\theta }}}{{{{360}^{\rm{\circ}}}}} \times \pi {r^2}\end{align}\)

(ii) Area of the segment = Area of the sector - Area of the corresponding triangle

From the figure,it is observed that 

Area of the designed region = \(2\times\)Area of the segment of the quadrant of radius \(\rm8\,cm\) = \(2\times\)[Area of the quadrant - Area of the right triangle]

Area of the quadrant

\[\begin{align} = \frac{{{{90}^ \circ }}}{{{{360}^ \circ }}} \times \pi {r^2} = \frac{1}{4}\pi {r^2}\end{align}\]

Area of the right triangle\(\begin{align} = \frac{1}{2} \times base \times height\end{align}\)

 

Steps:

From the figure, it is observed that

Area of the designed region \(=2\times\)Area of the segment of the quadrant of radius \(\rm{}8\,cm\)

\(=2\times\)[Area of the quadrant - Area of the right triangle]

\[\begin{align}&= 2 \times \left[ {\frac{1}{4}\pi {r^2} - \frac{1}{2} \times base \times height} \right]\\&= 2 \times \left[ {\frac{1}{4} \times \frac{{22}}{7} \times 8cm \times 8cm - \frac{1}{2} \times 8cm \times 8cm} \right]\\&= 2 \times \left[ {\frac{{352}}{7}c{m^2} - 32c{m^2}} \right]\\&= 2 \times \left[ {\frac{{352 - 224}}{7}c{m^2}} \right]\\&= 2 \times \frac{{128}}{7}c{m^2}\\&= \frac{{256}}{7}c{m^2}\end{align}\]