Question

A central angle$\theta$ in a circle of radius 9 m is subtended by an arc of length 14 m. find the measure of$\theta$ in degrees and radians.

Short answer

The central angle$\theta$ is${89.17}^{\circ }\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\frac{14}{9}\text{radian}$

Step-by-step solution

Step 1. Given information

The given radius of $\text{circle}=9\text{m}$and the length of the arc is 14 m.

Step 2. Explanation

We will use the formula of arc length,

$s=r\theta$

Step 3. Calculation

$\begin{array}{l}\text{Thelengthof\hspace{0.17em}the\hspace{0.17em}arc\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}s=r\theta \\ \theta =\frac{s}{r}\\ s=\frac{14}{9}\text{\hspace{0.17em}radian}\end{array}$

Now to convert central angle in degrees, multiply by $\frac{180}{\pi }$

$\begin{array}{l}\theta =\frac{14}{9}.\frac{180}{\pi }\\ \theta =\frac{280}{\pi }\\ \theta ={89.17}^{\circ }\end{array}$