Question

a circle has a circumference of 6. it has an arc of length $\frac{1}{3}$. what is the central angle of the arc, in degrees?

Explanation:

Step1: Set up the proportion

The ratio of the arc - length to the circumference is equal to the ratio of the central angle (in degrees) to 360 degrees. Let the central angle be $\theta$. So we have the proportion $\frac{\text{arc length}}{\text{circumference}}=\frac{\theta}{360}$.

Step2: Substitute the given values

We know that the arc length is $\frac{1}{3}$ and the circumference is 6. Substituting these values into the proportion, we get $\frac{\frac{1}{3}}{6}=\frac{\theta}{360}$.
First, simplify $\frac{\frac{1}{3}}{6}=\frac{1}{3}\div6=\frac{1}{3}\times\frac{1}{6}=\frac{1}{18}$. So our equation becomes $\frac{1}{18}=\frac{\theta}{360}$.

Step3: Solve for $\theta$

Cross - multiply: $18\theta = 360$. Then $\theta=\frac{360}{18}=20$.

Answer:

20