Question 192·Medium·Geometry and Trigonometry

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In a circle with radius $10$ centimeters, the length of arc $AB$ is $8\pi$ centimeters. What is the measure of the central angle that intercepts arc $AB$ , in degrees?

For arc length questions, first decide whether to use the formula $s = r\theta$ (with $\theta$ in radians) or the proportion $\dfrac{\text{arc length}}{2\pi r} = \dfrac{\text{central angle}}{360}$ . Plug in the radius and arc length, solve carefully for the angle (in radians or as a fraction of the full circle), and if you use radians, always multiply by $180/\pi$ to convert to degrees. Keep track of fractions and simplify step by step to avoid small arithmetic mistakes that lead to close but incorrect answer choices.

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Connect arc length to angle size

Ask yourself: how are arc length, the circle’s radius, and the central angle related? Think of a formula that uses all three.

Use a formula or a proportion

You can either use $s = r\theta$ with $\theta$ in radians, or use a proportion with the full circumference: $\dfrac{\text{arc length}}{\text{circumference}} = \dfrac{\text{central angle}}{360}$ .

Set up the equation with the given numbers

The radius is 10, so the circumference is $2\pi(10)$ . Plug $8\pi$ and $10$ into your chosen formula or proportion, then solve step by step for the angle. If you used radians, remember to convert to degrees at the end.

Desmos Guide

Compute the central angle directly

In Desmos, type the expression (8*pi/10)*(180/pi). The value that Desmos returns is the measure of the central angle in degrees.

Step-by-step Explanation

Relate arc length, radius, and central angle

For a circle, the relationship between arc length $s$ , radius $r$ , and central angle $\theta$ (in radians) is

s = r\theta

Here, $s = 8\pi$ centimeters and $r = 10$ centimeters. We first solve for $\theta$ in radians.

Solve for the central angle in radians

Use $s = r\theta$ and plug in the given values:

8\pi = 10\theta

Divide both sides by 10:

\theta = \frac{8\pi}{10} = \frac{4\pi}{5}

So the central angle is $\tfrac{4\pi}{5}$ radians. Now convert this to degrees.

Convert the angle from radians to degrees and choose the answer

To convert radians to degrees, multiply by $\dfrac{180}{\pi}$ :

\theta_{\text{degrees}} = \frac{4\pi}{5} \cdot \frac{180}{\pi}

The $\pi$ cancels, and you get

\theta_{\text{degrees}} = \frac{4 \cdot 180}{5} = \frac{720}{5} = 144

So the measure of the central angle is $144$ degrees, which corresponds to choice B.

Question 192·Medium·Geometry and Trigonometry

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation