Question

a central angle θ in a circle of radius 6 m is subtended by an arc of length 7 m. find the measure of θ in degrees. (round your answer to one decimal place.) θ =

find the measure of θ in radians. (round your answer to two decimal places.) θ = rad

Explanation:

Step1: Recall arc - length formula in radians

The formula for the length of an arc $s$ of a circle with radius $r$ and central - angle $\theta$ (in radians) is $s = r\theta$. We are given $s = 7$ m and $r = 6$ m. Solving for $\theta$ (in radians), we get $\theta=\frac{s}{r}$.
$\theta=\frac{7}{6}\approx1.17$ rad

Step2: Convert radians to degrees

The conversion formula from radians to degrees is $\theta_{degrees}=\theta_{radians}\times\frac{180^{\circ}}{\pi}$. Substitute $\theta_{radians}=\frac{7}{6}$ into the formula.
$\theta=\frac{7}{6}\times\frac{180^{\circ}}{\pi}=\frac{7\times180^{\circ}}{6\pi}=\frac{210^{\circ}}{\pi}\approx66.8^{\circ}$

Answer:

$\theta = 66.8^{\circ}$
$\theta = 1.17$ rad