Question

If the area of the below shape is $18\pi {\text{cm}}^2,$ then the value of $r$ is cm

Answer 1

Given shape is a semicircle as it formed by cutting a whole circle along a diameter line.

Here, $r$ is the distance from the center of the semicircle to it's circumference, so $r$ is the radius of the semicircle.

Given, Area of the semicircle $=18\times \pi {\text{cm}}^2$
We know: Area of semicircle $=\frac{1}{2}\times$ Area of the circle
$\therefore 18\pi =\frac{1}{2}\times \pi r^2$
Multiply both sides by 2
$\Rightarrow 18\pi \times 2=\frac{1}{2}\times \pi r^2\times 2$
$\Rightarrow 36\pi =\pi r^2$
Divide both sides by $\pi$
$\Rightarrow \frac{36\pi }{\pi }=\frac{\pi r^2}{\pi }$
$\Rightarrow 36=r^2$
Taking square root both sides
$\Rightarrow \sqrt{36}=\sqrt{r^2}$
$\Rightarrow \sqrt{6\times 6}=\sqrt{r^2}$
$\Rightarrow \pm 6=r$
$\Rightarrow r=6(\because \text{The radius of a semicircle cannot be negative}$
$\therefore r=-6\text{is rejected})$

Hence, the the value of $r$ is $6$ cm.