Question

If a semicircle has a circumference of $(8+4\pi )$ m, then its:

• A. $\text{radius = 2 m}$
• B. $\text{radius = 4 m}$
• C. $\text{area = 8}\pi \text{m}{}^2$
• D. $\text{area = 16}\pi \text{m}{}^2$

Answer 1

Answer: (B), (C)

$\text{area = 8}\pi \text{m}{}^2$

A semicircle is a half circle, formed by cutting a whole circle along a diameter line.

Step 1: From the above figure, the circumference of a semicircle
$=\text{diameter + half of the circumference of a circle}$ $=d+\frac{2\pi r}{2}$
$=2r+\pi r(\because d=2r)$

Step 2: Given, the circumference of the semicircle is $(8+4\pi )$ m.

$\therefore 2r+\pi r=8+4\pi$
Taking $r$ and $4$ common from L.H.S. and R.H.S. respectively
$\Rightarrow r(2+\pi )=4(2+\pi )$
Cancel out $(2+\pi )$ from both sides
$\Rightarrow r=4$ m

Step 3: $\text{Now, area of the semicircle}=\frac{1}{2}\times \text{Area of a circle}$
$=\frac{1}{2}\times \pi r^2$
$=\frac{1}{2}\times \pi \times 4^2$
$=\frac{1}{2}\times \pi \times 16$
$=8\pi {\text{m}}^2$

Hence, options (b.) and (c.) are correct choices.