McGraw-Hill Education GRE 2019

Sector Area and Arc Length

Sector area represents the area of a piece of a circle. Arc length represents the

length of a piece of a circle’s circumference.

A

B

Sector AB

A

B

Keep in mind the following:

■ To determine sector area, you will need to know the area of the circle and

what fraction the given piece is of the entire circle’s area.

■ To determine arc length, you will need to know the circumference of the

circle and what fraction the arc length is of the entire circumference.

If you are told what fraction the arc length or sector area is of a given circle, the

question is pretty straightforward. Let’s say a circle has an area of 16π. If you are

asked to determine the area of^14 of this circle, just divide the area by 4 and arrive at

4π. Likewise, let’s say a circle has a circumference of 8π. If you are asked to calculate

1

4 of the circumference, just divide the circumference by 4, and arrive at 2π.

Unfortunately, a real GRE question will not make things so simple. Instead of

telling you what fraction the sector is of the entire circle, a typical question will

expect you to infer this information from the central angle of the sector or arc. The

central angle is simply the angle formed by two intersecting radii. For example,

when you formed a quarter circle in the preceding example, the central angle was

90 degrees: A

B C

∠ABC = Central angle

From the preceding example, you can infer the following two relationships:

central angle

360 =

sector area

circle area

central angle

360 =

arc length

circumference

Why do these relationships hold? Because the central angle is simply a fraction of the

entire circle’s angle measurement. Since the angles around a circle’s center measure

360, a central angle of 60 degrees means the sector area is 36060 =^16 of the circle’s

area. Similarly, if the central angle is 40, then the arc length is simply 36040 =^19 of the

410 PART 4 ■ MATH REVIEW

04-GRE-Test-2018_313-462.indd 410 12/05/17 12:05 pm