Question

How do you find the radius of each circle if the area of the sector is $28$ and the angle is $150$ degrees?

Answer 1

Find the radius of the circle:

The area of the sector is obtained by $A=\frac{\theta }{360°}\times {\mathrm{\pi r}}^2$,

where $\theta$ is the angle and $r$ is the radius.

Put $A=28$ and $\theta =150°$ in $A=\frac{\theta }{360°}\times {\mathrm{\pi r}}^2$ and simplify to obtain the radius $r$.

$\begin{array}{rcl}28& =& \frac{150°}{360°}\times \left(\frac{22}{7}\right)r^2\\ & \Rightarrow & 28=\frac{5}{12}\times \left(\frac{22}{7}\right)r^2\\ & \Rightarrow & 28=\frac{110}{84}{r}^2\\ & \Rightarrow & \frac{28\times 84}{110}=r^2\left[\mathrm{Multiply}\mathrm{both}\mathrm{sides}\mathrm{by}\frac{84}{110}\right]\\ & \Rightarrow & 21.38=r^2\\ & \Rightarrow & \sqrt{21.38}=\sqrt{r^2}\left[\mathrm{Taking}\mathrm{square}\mathrm{root}\mathrm{of}\mathrm{both}\mathrm{sides}\right]\\ & \Rightarrow & 4.62=r\\ \therefore r& =& 4.62\end{array}$

Hence, the radius is $4.624\mathrm{unit}$.