Question

the radius of a circle is 5 centimeters. what is the angle measure of an arc 3π centimeters long?
t=3π cm
r=5 cm
give the exact answer in simplest form.
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π
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work it out

Explanation:

Step1: Calculate the circumference of the circle

The formula for the circumference of a circle is \(C = 2\pi r\). Given \(r = 5\) cm, then \(C=2\pi\times5 = 10\pi\) cm.

Step2: Set up a proportion to find the angle measure

Let the angle measure be \(x\) degrees. We know that the ratio of the arc length \(l\) to the circumference \(C\) is equal to the ratio of the central - angle measure \(x\) to \(360^{\circ}\). That is \(\frac{l}{C}=\frac{x}{360}\). Substitute \(l = 3\pi\) and \(C = 10\pi\) into the proportion: \(\frac{3\pi}{10\pi}=\frac{x}{360}\). Since \(\pi\) cancels out (\(\frac{3\pi}{10\pi}=\frac{3}{10}\)), we have \(\frac{3}{10}=\frac{x}{360}\).

Step3: Solve for \(x\)

Cross - multiply: \(10x=3\times360\). Then \(10x = 1080\). Divide both sides by 10: \(x=\frac{1080}{10}=108\).

Answer:

\(108\)