# Ex.12.3 Q7 Areas Related to Circles Solution - NCERT Maths Class 10

## Question

In the given figure, \(ABCD\) is a square of side \(\text{14 cm.}\) With Centers \(A, B, C\) and \(D,\) four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region.

## Text Solution

**What is known?**

\(ABCD\) is a square of side \(\text{= 14 cm.}\)

With centers\( A, B, C, D\) four circles are drawn such that each circle touches externally \(2\) of the remaining \(3\) circles.

**What is unknown?**

Area of the shaded region.

**Reasoning:**

Since the circles are touching each other externally,visually it is clear that

Radius of each circle \(\begin{align} {r} = \frac{1}{2}\,\, \times \end{align} \)(side of square)

Also, \(ABCD\) being a square all angles are of measure \({90^ \circ },\,\,\, \)

Therefore,all sector are equal as they have same radii and angle.

\(\therefore\;\) Angle of each sector which is part of the square \(\left( {\theta } \right) = {90^ \circ }\)

\(\therefore\;\)\(\text{Area of each sector}\)\[\begin{align}&= \frac{\theta }{{{{360}^\circ }}} \times \pi {r^2}\\&= \frac{{{{90}^\circ }}}{{{{360}^\circ }}} \times \pi {r^2}\\&= \frac{{\pi {r^2}}}{4}\end{align}\]

From the figure it is clear that:

Area of shaded region \(=\) Area of square \(– \) Area of \(4\) sectors

\[\begin{align} &= {\left( {{\text{side}}} \right)^2} - 4 \times {\text{Area of each sector}}\\ &= {\left( {14} \right)^2} - 4 \times \frac{{\pi {r^2}}}{4}\\ &= {\left( {14} \right)^2} - \pi {r^2}\end{align}\]

**Steps:**

Area of each of the \(4\) sectors is equal as each sector subtends an angle of \({90^ \circ }\) at the centre of a circle with radius \(\text{= 7 cm}\)

\(\therefore\) \(\text{Area of each sector}\)\[\begin{align}&= \frac{\theta }{{{{360}^\circ }}} \times \pi {r^2}\\ &= \frac{{{{90}^\circ }}}{{{{360}^\circ }}} \times \pi {(7)^2}\\ &= \frac{1}{4} \times \frac{{22}}{7} \times 7 \times 7\\ &= \frac{{77}}{2}{\text{c}}{{\text{m}}^2}\end{align}\]

Area of shaded region \(=\) Area of square \(- \,4 \;\times\) Area of each sector

\[\begin{align}&= {(14)^2} - 4 \times \frac{{77}}{2}\\ &= 196 - 154\\ &= 42\,\,{\text{c}}{{\text{m}}^2}\end{align}\]