Advanced Geometry
Circles
Lesson 4

There are many real-world applications which can be solved more easily using an angle measure other than the degree.
This other unit is called the radian.

How do Radians relate to Degrees?
For a circle with a radius of 1 unit, C r
2
 
In degrees, the measure of a full circle is 360 °.
So, 2
 
360 .
π radians = 180 °

π radians = 180 °
Angles expressed in radians are written in terms of

.
180
1 radian = degrees

1 degree = radians
180

Examples:
Change 115 ° to radian measure in terms of 
.
23

Change radians to degree measure to the nearest hundredth.
8


36
157.5

Change 5 radians to degree measure to the nearest hundredth.

Arc Length length of a circular arc central angle r
 s s
 r

θ must be measured in radians.
 s

Examples: 
Given a central angle of , find the length of its
4 intercepted arc in a circle of radius 3 inches. Round to the nearest hundredth.

Examples:
Given a central angle of 125 °, find the length of its intercepted arc in a circle of diameter 14 centimeters.
Round to the nearest hundredth.

Examples:
An arc is 14.2 centimeters long and is intercepted by an central angle of 60 °. What is the radius of the circle to the nearest hundredth?

Definition – a region bounded by a central angle and the intercepted arc


1
A r
2
2

 r
θ must be measured in radians.

Example:
Find the total area of the yellow sector to the nearest hundredth.
in 2
Find the probability that a point chosen at random lies in the yellow shaded area.