McGraw-Hill Education GRE 2019

circumference. So how will you use the central angle to determine sector area? Let’s

look at an example.

If a circle has an area of 90π, what is the area of a sector with a central angle

of 60 degrees?

1. Determine what fraction the sector is of the entire circle. Use the formula:
   central angle 360 = sector areacircle area
   In this case, 36060 =^16. Thus the sector area is^16 of the circle’s entire area.


2. Multiply the area of the circle by^16 : 90π ×^16 = 15π.
   In the example, you used the central angle to determine the sector area, but you
   can also use sector area or arc length to determine the central angle.

A sector has an arc length of 9π and an area of 100π. What is the central

angle of the sector?

SOLUTION: Use the formula central angle/360 = arc length/circumference. You

know the arc length, so to determine the central angle, you need to determine

the circumference, and then solve the proportion. You can use the area to

determine the radius: 100π= πr^2. Thus r = 10. If r = 10, then the diameter = 20,

and the circumference = 20π. Now you can plug these values into the original

formula: substitute 20π for the circumference and 9π for the arc-length:

central angle 360 = circumferencearc length

↓

central angle 360 = 20π9π

central angle 360 = 209

central angle × 20 = 360 × 9

central angle = 162

Triangle in a Semicircle

If a triangle is inscribed in a semicircle, two properties follow:

1. The triangle is a right triangle.


2. The diameter of the circle = the triangle’s hypotenuse.
   B

A C

CHAPTER 13 ■ GEOMETRY 411

04-GRE-Test-2018_313-462.indd 411 12/05/17 12:05 pm