Question 92·Medium·Math

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A circle has radius $12\text{ centimeters}$ . A sector of the circle has a central angle of $150^{\circ}$ . What is the area, in square centimeters, of the sector?

For sector area questions, immediately write the formula $A = \frac{\theta}{360} \pi r^2$ or think of it as “fraction of circle’s angle times area of full circle.” Plug in the given radius and central angle, simplify the fraction $\theta/360$ first, then multiply by $r^2$ and $\pi$ . Always double-check that you used the radius (not the diameter) and that the angle is treated as a fraction of 360°, which keeps the arithmetic quick and reduces mistakes.

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Relate the sector to the whole circle

Think about how the area of a sector compares to the area of the whole circle. How does the central angle of the sector (in degrees) help you find what fraction of the circle it represents?

Use the sector area formula

Recall the formula for the area of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees): it involves the fraction $\theta/360$ multiplied by the area of the full circle $\pi r^2$ .

Plug in and simplify carefully

Substitute $r = 12$ and $\theta = 150$ into the sector area formula. Simplify $12^2$ and reduce the fraction $150/360$ before multiplying to keep the arithmetic easy.

Desmos Guide

Use Desmos to compute the sector area

In Desmos, type the expression (150/360)*pi*12^2 and press Enter. Compare the simplified symbolic result shown by Desmos to the answer choices; the matching expression is the correct choice.

Step-by-step Explanation

Write the formula for the area of a sector

For a circle with radius $r$ and a sector with central angle $\theta$ degrees, the area $A$ of the sector is

A = \frac{\theta}{360} \cdot \pi r^2

This formula says: take the fraction of the full circle ( $\theta/360$ ) and multiply by the area of the whole circle ( $\pi r^2$ ).

Substitute the given radius and angle

Here, the radius is $r=12$ centimeters and the central angle is $\theta = 150^\circ$ .

Substitute these into the formula:

A = \frac{150}{360} \cdot \pi \cdot 12^2

Calculate $12^2$ :

12^2 = 144

So the expression becomes

A = \frac{150}{360} \cdot 144\pi

Simplify the fraction and multiply

First reduce the fraction $\frac{150}{360}$ :

\frac{150}{360} = \frac{15}{36} = \frac{5}{12}

Now multiply this by $144\pi$ :

A = \frac{5}{12} \cdot 144\pi

Compute $\frac{144}{12} = 12$ , so

A = 5 \cdot 12\pi = 60\pi

So the area of the sector is $60\pi$ square centimeters, which corresponds to choice C.

Question 92·Medium·Math

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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