In circle , the radius is . If the measure of minor arc is , what is the length, in centimeters, of arc ?

Solution available with step-by-step explanation

SAT Circles Question 21 | Free SAT Prep | Aniko

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Question 21·Medium·Circles

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In circle $O$ , the radius is $10\text{ cm}$ . If the measure of minor arc $AB$ is $72^\circ$ , what is the length, in centimeters, of arc $AB$ ?

For arc-length questions, immediately recall the key idea: arc length equals (central angle ÷ 360) times the circle’s circumference. First compute the full circumference using $2\pi r$ , then find the fraction of the circle given by the central angle, simplify that fraction if possible, and multiply. Finally, match your expression (often a multiple of $\pi$ ) directly to the answer choices, watching out for errors in simplifying the angle fraction or mixing up minor and major arcs.

Hints

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Relate the arc to the whole circle

Think about how arc length is connected to the central angle. How is the central angle of an arc related to $360^\circ$ for a full circle?

Find the circle’s circumference first

Use the radius of $10$ cm to find the full circumference of the circle using $C = 2\pi r$ .

Use a fraction of the circumference

Once you know the circumference, multiply it by the fraction $\frac{\text{central angle}}{360}$ to get the arc length. What is $\frac{72}{360}$ as a simplified fraction?

Desmos Guide

Compute the arc length directly

In Desmos, type the expression (72/360)*2*pi*10 to represent $\frac{72}{360} \cdot 2\pi \cdot 10$ . Look at the result Desmos gives, and then compare that value to each answer choice (you can also type each choice, like 4*pi, 5*pi, etc., into Desmos to see which one matches).

Step-by-step Explanation

Use the arc length formula

For a circle, the length of an arc is given by:

\text{arc length} = \frac{\text{central angle}}{360^\circ} \cdot \text{circumference of the circle}

Here, the central angle for minor arc $AB$ is $72^\circ$ , and the radius is $10$ cm.

Find the circle’s circumference

Use the circumference formula $C = 2\pi r$ .

With radius $r = 10$ cm:

C = 2\pi(10) = 20\pi

So the entire circle has circumference $20\pi$ cm.

Find what fraction of the circle the arc represents

Compare the central angle to a full circle of $360^\circ$ :

\frac{72}{360} = \frac{1}{5}

So minor arc $AB$ is $\frac15$ of the full circle.

Compute the arc length and match the choice

Multiply this fraction by the circumference:

\text{arc length of } AB = \frac15 \cdot 20\pi = 4\pi

So the length of arc $AB$ is $4\pi$ centimeters, which corresponds to choice D.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation