Question

If a wire is bent into the shape of a square, then the area enclosed by the square is $81c{m}^2$. When the same wire is bent into a semi-circular shape, then the area enclosed by the semi circle will be

• A. $81c{m}^2$
• B. $44c{m}^2$
• C. $77c{m}^2$
• D. $154c{m}^2$

Answer 1

The correct option is C $77c{m}^2$

Let the side of the square be $a$ cm.

Now, area of square$=a^2$

$\Rightarrow 81=a^2\Rightarrow a=9cm$

Now, length of wire$=4\times a$

$=4\times 9$

$=36cm$

Perimeter of semicircular wire$=\pi r+2r$

Also, perimeter of Semicircular wire$=$Length of wire

$\therefore \pi r+2r=36$

$\Rightarrow r(\frac{22}{7}+2)=36$

$\Rightarrow \frac{36}{7}\times r=36$

$\Rightarrow r=7cm$

Area of semicircular shape$=\frac{1}{2}\times \pi r^2$

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

$=77{cm}^2$