Pre-Calculus
Notes
Name: ___________
Date: ____________
Lesson 4.1B: Arc Lengths and Sector Areas
Learning Targets:
B:
C:
D:
E:
O:
Convert an angle measure from DMS to degree notation or from degree to DMS notation.
Calculate the length of a given arc.
Calculate the area of a given sector.
Calculate linear and angular speed.
Use angles to model and solve real-life problems.
Vocabulary:
Arc length
Sector area
Linear velocity
Angular velocity
Angle Measures in D M S Form:
1.
Convert each angle measure to decimal degree form.
2.
a.
24510
b.
40816 20
Convert each angle measure to D M S form.
a.
-345.12 
b.
0.7865 
Arc Length:
Consider a circle with radius r and a central angle  measured in radians.
Then the length, s, of the arc intercepted by  is given by:
s = __________
3.
Find the length of the arc that subtends a central angle with measure 120  in a
circle with radius 5 inches.
Sector Area:
Consider a circle with radius r and a central angle  measured in radians.
Then the area, A, of a sector intercepted by  is given by:
A = __________
4.
Find the area of a sector subtended by a central angle with measure 150  in a
circle with radius 6 inches.
Linear and Angular Velocity:
Consider a particle moving at a constant speed along a circular arc of radius r.
If s is the length of the arc traveled in time t, then the linear speed, v, of the
particle is
𝑣=
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
=
𝑡𝑖𝑚𝑒
=
If  is the angle, measured in radians, corresponding to the arc length s, then the
angular speed,  , of the particle is
𝜔=
5.
6.
𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒
=
𝑡𝑖𝑚𝑒
A carousel with a 50-foot diameter makes 4 revolutions per minute.
a.
Find the linear speed of the platform rim of the carousel.
b.
Find the angular speed of the carousel in radians per minute.
An automobile is traveling at 65 mph. If its tires have a radius of 15 inches,
at what rate, in rpms, are the tires spinning?