Question

Find the area of the shaded region in the given figure where circles are drawn with centres $A,B,C$ and $D$ intersected in pairs at mid-points $P,Q,R$ and $S$ of the sides $AB,BC,CD$ and $DA$ respectively of a square $ABCD$ of side $14cm$. (Use $\pi =\frac{22}{7})$.

• A. $22c{m}^2$
• B. $36c{m}^2$
• C. $42c{m}^2$
• D. $157.5c{m}^2$

Answer 1

Answer: (C)

Given, $ABCD$ is a square of side $14cm$

$P,Q,R$ and $S$ are the midpoints of sides $AB,BC,CD$ and $AD$ respectively.

In figure, each non-shaded region is a quadrant. [Since, $ABCD$ is a square.]

$\Rightarrow$ Area of each non shaded region $=$ Area of a quadrant

$\Rightarrow$ Area of each non shaded region $=\frac{1}{4}\times \pi r^2$ [Since, area of quadrant is equal to $\frac{1}{4}\text{th}$ of the area of circle.]

Now,

Area of shaded region $=$ Area of square $-4\times$ Area of quadrant

$=a^2-4\times \frac{1}{4}\pi r^2$

$=(14cm{)}^2-\pi (7cm{)}^2$

$=196c{m}^2-\frac{22}{7}\times 49c{m}^2$ [$\because \pi =\frac{22}{7}$]

$=196c{m}^2-154c{m}^2$

$=42c{m}^2$

Hence, Option $C$ is correct.