SAT Circles Question 35 | Free SAT Prep | Aniko

00:00

Question 35·Medium·Circles

0 / 50

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.

Hints

Click me!

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.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation