Question 140·Medium·Geometry and Trigonometry

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A circle has a radius of 6 centimeters. What is the length, in centimeters, of the arc that subtends a central angle of 120 degrees?

For circle arc-length problems on the SAT, immediately recall that arc length equals the fraction of the circle’s angle over 360 multiplied by the full circumference, $s = \dfrac{\theta}{360} \cdot 2\pi r$ . Plug in quickly, simplify the angle fraction first (like $120/360 = 1/3$ ), then multiply the remaining numbers and keep $\pi$ symbolic unless the question tells you to approximate. This avoids mistakes with degrees and saves time on calculations.

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Think of the arc as part of the full circle

How can you express the arc length as a fraction of the full circumference of the circle?

Recall the circumference formula

What is the formula for the circumference of a circle in terms of its radius $r$ ?

Use the angle fraction

What fraction of a full circle (360 degrees) is a 120-degree central angle? Multiply that fraction by the full circumference.

Desmos Guide

Compute the arc length directly

In Desmos, type the expression (120/360)*2*pi*6. This expression represents the fraction of the full circumference that corresponds to 120 degrees. The numerical value that Desmos outputs is the length of the arc.

Step-by-step Explanation

Relate arc length to circumference

An arc with central angle $\theta$ degrees is $\dfrac{\theta}{360}$ of the full circle. The full circumference of a circle with radius $r$ is $2\pi r$ , so the arc length $s$ is:

s = \frac{\theta}{360} \cdot 2\pi r

Here, $\theta = 120$ and $r = 6$ .

Substitute the given values

Substitute $\theta = 120$ and $r = 6$ into the formula:

s = \frac{120}{360} \cdot 2\pi \cdot 6

Now simplify the fraction $\dfrac{120}{360}$ and the multiplication.

Simplify to find the arc length

First, simplify the fraction:

\frac{120}{360} = \frac{1}{3}

Then compute the product $2 \cdot 6 = 12$ , so:

s = \frac{1}{3} \cdot 12\pi = 4\pi

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

Question 140·Medium·Geometry and Trigonometry

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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