Question

the diameter of a circle is 8 inches. what is the angle measure of an arc 3π inches long?
t=3π in

d=8 in

give the exact answer in simplest form.

°

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work it out

Explanation:

Step1: Calculate the circumference of the circle

The formula for the circumference of a circle is \(C=\pi d\). Given \(d = 8\) inches, then \(C=\pi\times8=8\pi\) inches.

Step2: Set up the proportion for the arc length

The proportion for the arc length \(l\) and the central angle \(\theta\) (in degrees) is \(\frac{l}{C}=\frac{\theta}{360}\). We know \(l = 3\pi\) and \(C=8\pi\). Substitute these values into the proportion: \(\frac{3\pi}{8\pi}=\frac{\theta}{360}\).

Step3: Solve for \(\theta\)

Since \(\frac{3\pi}{8\pi}=\frac{3}{8}\), our equation becomes \(\frac{3}{8}=\frac{\theta}{360}\). Cross - multiply: \(8\theta=3\times360\). Then \(8\theta = 1080\). Divide both sides by 8: \(\theta=\frac{1080}{8}=135\).

Answer:

\(135\)