Question

A circle has a circumference of \blue{6}6start color #6495ed, 6, end color #6495ed. It has an arc of length \dfrac{1}{3} 3 1 ​ start fraction, 1, divided by, 3, end fraction. What is the central angle of the arc, in degrees?

Answer 1

`\textit{circumference of a circle}\\\\ C=2\pi r ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ C=6 \end{cases}\implies 6=2\pi r\implies \cfrac{6}{2\pi }=r\implies \cfrac{3}{\pi }=r \\\\[-0.35em] ~\dotfill`

`\textit{arc's length}\\\\ s = \cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=(3)/(\pi )\\[1em] s=(1)/(3) \end{cases}\implies \cfrac{1}{3}=\cfrac{~~ \theta \cdot \pi \cdot (3 )/(\pi ) ~~}{180}\implies \cfrac{1}{3}=\cfrac{\theta }{60} \\\\\\ 60=3\theta \implies \cfrac{60}{3}=\theta \implies \boxed{20=\theta}`