Question

Find the area of the shaded portions in the figure, where ABCD is a square of side 7 cm.

Answer 1

Let us mark the unshaded portions by$I,II,IIIandIV$ as shown in the figure.

Let$P,Q,R$ and$S$ be the mid points of $AB,BC,CDandDA$respectively.

Side of the square, $a=7cm$

Radius of the semicircle, $r=\frac{7}{2}cm$

Area of $I+$ Area of $III=$ Area of a square ABCD - Area of two semicircles with centres$P$ and$R$

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

$=7\times 7-2\times \frac{1}{2}\times \frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}$

$\therefore$ Area of$I$+ Area of$III$$=(49-\frac{77}{2})c{m}^2=\frac{21}{2}c{m}^2$.

Similarly, we have

Area of$II$ + Area of $IV=$$(49-\frac{77}{2})c{m}^2=\frac{21}{2}c{m}^2$.

Area of the shaded portions = Area of the square $ABCD$ - (Area of I + Area of II + Area of III + Area of IV)

$=49-(\frac{21}{2}+\frac{21}{2})$

$=49-21=28c{m}^2$

$\therefore$ Area of the shaded portions $=28c{m}^2$