Question

Find the area of the shaded region.

• A. $(180-2\pi )c{m}^2$.
• B. $(180-8\pi )c{m}^2$.
• C. $(120-8\pi )c{m}^2$.
• D. $(180-6\pi )c{m}^2$.

Answer 1

The correct option is B

$(180-8\pi )c{m}^2$.

Join JK, KL, LM and MJ

There are four equal semi - circles and LMJK formed is a square.

$\therefore$ FH = 14 – ( 3 +3 ) = 8cm

So, the side of the square should be 4 cm and radius of semi - circles = 2 cm each.

$\therefore$ Area of square JKLM $=(4{)}^2=16c{m}^2$

Area of semi - circle HJM $=\frac{\pi r^2}{2}$

$\frac{\pi \times (2{)}^2}{2}=2\pi c{m}^2$

$\therefore$ Area of four semi-circle $=4\times 2\pi c{m}^2=8\pi c{m}^2$

Now, area of square ABCD $=(14{)}^2=196c{m}^2$

$\therefore$ Area of shaded region = Area of square ABCD - [Area of four semi-circles + Area of square JKLM]

$=196-[8\pi +16]=196-16-8\pi$

$=(180-8\pi )c{m}^2$

Hence, the area of the shaded region is $(180-8\pi )c{m}^2$.