Question

An arc of length $100m$ subtends a central angle $\theta$ in a circle of radius $50\mathrm{m}$. Find the measure of $\theta$ in degrees and in radians.

Answer 1

Find the measure of $\theta$ in degrees and in radians.

We know that in a circle of radius $r$, if an arc length is $l$ subtends an $\angle \theta$ at the center, then $\mathrm{\theta }=\frac{\mathrm{l}}{\mathrm{r}}$.

It is given $\mathrm{l}=100\mathrm{m}$, $\mathrm{r}=50\mathrm{m}$

Thus $$\begin{gathered} \mathrm{\theta }=\frac{100}{50}\mathrm{radian} \\ =2\mathrm{radian} \end{gathered}$$

convert radian into degree

$$\begin{gathered} \mathrm{\theta }=2\times \frac{180}{\mathrm{\pi }}\mathrm{degree} \\ =\frac{2\times 180}{{\displaystyle \frac{22}{7}}} \\ =\frac{2\times 180\times 7}{22} \\ =144.54° \end{gathered}$$

Therefore, the value of $\mathrm{\theta }$ is $2\mathrm{radian}\mathrm{or}144.54°$.