Objective: Students will be able to use angles to model and solve real-life problems.
Radian measure can be used to find the length of an arc of a sector of a
circle. Let’s look to see what this means.
FORMULA FOR ARC LENGTH:
Example 1: Find the length of an intercepted arc in a circle with radius 16 if the central angle
is
p
4
radians.
Example 2: Find the length of an intercepted arc in a circle with radius 12 if the central angle
is 150o.
Example 3: Find the central angle of the
7.7 cm
circle.
4 cm
The formula for arc length can be used to find the speed of a particle moving along a circular
path.
 Angular Displacement –
 Angular Velocity –
 Linear Velocity –
Example 4: If the arc length of a sector is 14 inches and the radius is 20 inches, what is the
radian measure of the central angle?
Example 5: Determine the angular displacement, in radians, of 4.5 revolutions.
Example 6: Determine the angular velocity of a particle if it completes 7.3 revolutions in 5
seconds.
Example 7: Determine the linear velocity of a point rotating at an angular velocity of 17π
radians per second at a distance of 5 centimeters from the center of the rotating object.
Example 8: A car’s wheels are 28 inches in diameter. How far (in miles) will the car travel if
its wheels revolve 10,000 times?
Example 9: To measure the speed of a current, scientist place a paddle wheel in the stream
and observe the rate at which it rotates. If the paddle wheel has a radius of 0.20 meters and
rotates at 100 rpm(revolutions per minute), find the speed of the current in meters per
second.
Example 10: Consider the Earth, which rotates on its axis once every 24 hours.
 This complete circle is __________ radians. Therefore, the angular velocity of the Earth’s
rotation is
 A ceiling fan rotates 30 times per minute, the angular velocity in radians per hours is
Example 11: Now, consider the linear velocity of both objects…

The radius of the Earth is approximately 3960 miles. The Earth is rotating at rate of π/12
radians per hour. What is the linear velocity in miles per hour?

The radius of the fan above is 2 feet. The fan is rotating at a rate of 3600π radians per
hour. What is the linear velocity in miles per hour?
https://www.youtube.com/watch?v=9QDKnQm_poI
Warm-Up
1) Memphis, TN is due north of New Orleans, LA. Find the distance between Memphis
(35o9’ north latitude) and New Orleans (29o57’ north latitude). Assume that the radius
of the earth is 4000 miles.
2) Boston is located 42o21’29” north of the equator. The earth rotates on an axis through
the poles. The distance from the axis to a point in Boston is shorter then the distance
from the axis to a point on the equator. Compute the linear speed of Boston.
Warm-Up
1) Memphis, TN is due north of New Orleans, LA. Find the distance between Memphis
(35o9’ north latitude) and New Orleans (29o57’ north latitude). Assume that the radius
of the earth is 4000 miles.
2) Boston is located 42o21’29” north of the equator. The earth rotates on an axis through
the poles. The distance from the axis to a point in Boston is shorter then the distance
from the axis to a point on the equator. Compute the linear speed of Boston.