Question 44·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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The area of a circle is $144\pi$ square inches. A sector of the circle has area $40\pi$ square inches.

Which choice is the length of the arc that bounds the sector?

A sector is a slice of a circle, and its arc is the curved edge. When you know the circle’s area and the sector’s area, find the radius, then use the sector’s area fraction to take the same fraction of the circumference. Don’t take a fraction of the radius: the arc is part of the distance around the circle.

Hints

Hint 1
The radius is the distance from the center to the edge. The whole circle’s area is $A=\pi r^2$ , so match that formula to the given area. What positive radius does it give?
Hint 2
A sector is a slice of a circle. It takes up the same fraction of the circle’s area and circumference. Divide the sector’s area by the whole circle’s area to find that fraction.
Hint 3
The circumference is the full distance around a circle, $2\pi r$ . The requested arc is only the sector’s share of that distance. What do you get when you multiply the circumference by the fraction you found?

Step-by-step

Use the sector’s fraction of the circle
Step 1
Turn the circle’s area into a radius equation
The radius $r$ runs from the center to the edge. A circle’s area is $A=\pi r^2$ , so the given whole-circle area gives $\pi r^2=144\pi$ .
Step 2
Find the positive radius
Type $\pi r_1^2\sim144\pi\{r_1>0\}$ in Desmos. The $\sim$ asks Desmos to solve for $r_1$ , and the restriction keeps the radius positive because a length can’t be negative. Under PARAMETERS, Desmos shows $r_1=12$ , so the radius is $12$ inches.
Step 3
Find the sector’s fraction
The sector is a slice bounded by two radii and an arc. Its share of the area is the same fraction as its share of the circumference. Divide its area by the whole area: $f=\frac{40\pi}{144\pi}$ . Type that fraction in Desmos. It shows $0.2777\ldots$ ; click the fraction button to see $\frac{5}{18}$ , so $f=\frac{5}{18}$ . Use that fraction of the circumference, not of the radius.
Step 4
Find the whole circumference
The circumference is the full distance around the circle. With radius $12$ inches, $C=2\pi r=2\pi(12)=24\pi\text{ inches}$ .
Step 5
Take the fraction that belongs to the arc
The arc is a length, so take $\frac{5}{18}$ of the circumference: $s=\frac{5}{18}(24\pi)=\left(\frac{5}{18}\cdot24\right)\pi$ . Type $\frac{5}{18}\cdot24$ in Desmos. It shows $6.666\ldots$ ; click the fraction button to get $\frac{20}{3}$ . Type $\frac{20}{3}\pi$ as the last line; Desmos shows about $20.944$ . The arc length is $\frac{20\pi}{3}$ inches. Choice B.

Question 44·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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