Question 108·Medium·Geometry and Trigonometry

0 / 200

A circle has a circumference of $24\pi$ centimeters. What is the area, in square centimeters, of a sector of the circle determined by a central angle that measures $60^{\circ}$ ?

For circle-sector problems on the SAT, first use the given circumference or area to find the radius with $C = 2\pi r$ or $A = \pi r^2$ . Then compute the full circle's area if needed, and finally multiply by the fraction of the circle represented by the central angle (angle over 360). Keeping this three-step structure in mind (radius → whole area → fraction of circle) helps you move quickly and avoid mixing up formulas or angle fractions.

Hints

Click me!

Relate circumference to radius

You know the circle's circumference. What formula connects circumference and radius, and how can you use it to find the radius?

Use the radius to get the full circle's area

Once you have the radius, apply the formula for the area of a circle. Do not think about the sector yet; just find the area of the entire circle.

Connect the central angle to a fraction of the circle

A full circle is $360^\circ$ . What fraction of the full circle is a $60^\circ$ sector? Use that fraction of the total area to get the sector's area.

Desmos Guide

Compute the radius from the circumference

In Desmos, type 24pi/(2pi) and note the value of the result; this is the radius of the circle.

Compute the area of the full circle

Using the radius from step 1, type pi*(result)^2 (or directly pi*(24pi/(2pi))^2) to get the area of the entire circle.

Compute the sector's area

Now multiply the full area by the fraction 60/360. In Desmos, you can type (60/360)*pi*(24pi/(2pi))^2. The numerical expression that Desmos returns is the area of the $60^\circ$ sector.

Step-by-step Explanation

Use the circumference to find the radius

The circumference of a circle is given by $C = 2\pi r$ .

We are told $C = 24\pi$ , so set up the equation:

2\pi r = 24\pi

Now solve for $r$ by dividing both sides by $2\pi$ :

r = \frac{24\pi}{2\pi} = 12

So the radius of the circle is 12 centimeters.

Find the area of the entire circle

The area of a circle is given by $A = \pi r^2$ .

Using $r = 12$ :

A = \pi (12)^2 = \pi \cdot 144 = 144\pi

So the area of the whole circle is $144\pi$ square centimeters.

Find the fraction of the circle represented by a 60° sector

A full circle has an angle of $360^\circ$ at its center.

A sector with a central angle of $60^\circ$ is this fraction of the full circle:

\frac{60}{360} = \frac{1}{6}

So the sector is $\tfrac{1}{6}$ of the entire circle.

Compute the area of the sector and match the choice

The area of a sector is the same fraction of the circle's area as its angle is of $360^\circ$ .

So the sector area is:

\text{Sector area} = \frac{1}{6} \times 144\pi = 24\pi

Thus, the area of the sector is $24\pi$ square centimeters, which corresponds to choice B.

Question 108·Medium·Geometry and Trigonometry

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation