The circumference of a circle is units. The area of a sector of the circle is square units. Which choice is the measure, in degrees, of the central angle of the sector?

Solution available with step-by-step explanation

SAT Circles Question 36 | Free SAT Prep | Aniko

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Question 36·Hard·Circles

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The circumference of a circle is $30\pi$ units. The area of a sector of the circle is $75\pi$ square units.

Which choice is the measure, in degrees, of the central angle of the sector?

When a sector problem gives different circle measures, connect them through the whole circle. Here, the circumference gives the radius, the radius gives the full area, and the area ratio gives the central-angle ratio. Breaking the problem into those linked steps makes the calculation straightforward.

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Start with the circumference

Use $2\pi r = 30\pi$ to find the radius before working with the area of the sector.

Use the area ratio

The sector takes up the same fraction of the full circle''s area as its central angle takes up of $360^\circ$ .

Desmos Guide

Find the radius and full area

In Desmos, enter r = 30*pi/(2*pi) and then A = pi*r^2. This gives the area of the entire circle.

Use the area fraction to find the angle

Enter (75*pi/A)*360. The result is the measure of the central angle and matches the correct answer choice.

Step-by-step Explanation

Find the radius and total area

Because

2\pi r=30\pi,

the radius is $r=15$ . So the area of the full circle is

\pi r^2 = 225\pi.

Find the fraction of the circle

The sector area is

\frac{75\pi}{225\pi}=\frac13

of the full area.

Convert the fraction to an angle

The sector''s central angle is the same fraction of $360^\circ$ :

\frac13\cdot 360^\circ = 120^\circ.

So the correct choice is $120$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation