5.1 Angles and Their Measures
Pre-Calculus Notes
Name ________________________________
Date ___________________
Period _____
EX: Application: Finding the Distance Between Two Cities
1. Find the distance between Erie, PA with a location of 42° 5' 25" N and Savannah, GA with a location
31°57’ N.
2. You are flying from Blacksburg, VA with a location of 42°5'25" N and Rio Di Janeiro, Brazil with a location
31°57'23"S. How far did you travel?
Area of Sector:
The area A of the sector of a circle of radius r formed
by a central angle of  radians is
A
1 2
r 
2
EX: 3. Find the area of the sector of a circle of radius 3.5 feet formed by an angle of 65. Round the answer to
the nearest thousandth.
4. Find the area of the sector of a circle of diameter 10 feet formed by an angle of 135. Round the answer to the
nearest thousandth.
Circular Motion:
Linear Speed:
v
s
t
Angular Speed:


t
EX: 5. An object is traveling around a circle with a radius of 5 cm. If in 25 seconds a central angle of
1
radian is
4
swept out, what is the angular speed of the object? What is its linear speed?
6. A phonograph record has a radius of 3 inches and revolves at 45 RPM Find the linear speed of the outside
edge of the record.
7. A car is traveling 60mph. The diameter of the wheels is 3 ft.
a) Find the number of revolutions per minute the wheels are rotating.
b) What’s the angular speed of the wheels in radians per minute.
Assume the hard drive on a computer is circular and rotates at 7200 revolutions per minute. What
is the angular velocity in radians per minute? What is the linear velocity in inches per minute of a
particle located 2 inches from the center of the hard drive? …. in miles per hour?
The angular velocity equals 7200 revolutions per minute =
The linear velocity is
.
To change this to mph we have
.
Minutes, feet, and inches all cancel and we have
.
A particle moves on a circular path with a linear velocity of 300 feet per second. If the particle
makes 3 revolutions per second, what is its angular velocity? What is the radius of the circle?
Its angular velocity equals 3 revolutions per second is
Using
have
.
we
.
i.
A Ferris Wheel rotates 3 times each minute. The passengers sit in seats that are 25 feet
from the center of the wheel. What is the angular velocity of the wheel in degrees per
minute and radians per minute? What is the linear velocity of the passengers in the seats?
3 revolutions per minute
is
.
The linear velocity is calculated from
.
This gives us
We can change this to miles per hour by multiplying as follows:
.
.
We can simplify this result by “canceling” the minutes and feet to get
.
ii.
An object is rotating on a circular path at 4 revolutions per minute. The linear velocityof
the object is 400 feet per minute. What is the radius of the circle and what is the
angular velocity of the rotating platform?
We first find the angular velocity: 4 revolutions per minute is
Second, we use
.
. This gives us
.
iii.
An object moves along a circular path of radius r. What is the effect on the linearvelocity if
the radius of this circle is doubled?
The linear velocity is
have:
doubled.
. If the linear velocity is recalculated using 2r as the new radius, we
. Therefore if the radius is doubled, the linear velocity is also
Warm Up:
1. An angle  is in __________ ____________ if its vertex is at the origin of a rectangular coordinate system
and its initial side coincides with the positive x-axis.
2. On a circle of radius r, a central angle of  radians subtends an arc of length s = _____; the area of the
sector formed by this angle  is A = _________
3. TRUE or FALSE? π = 180