Radians & Arc Length
1
MEASURING ANGLES
Degrees
Radians
A radian is the central angle that subtends an arc length of one radius.
What is relationship between radians & degrees?
360 = 2 radians
180radians
Radians & Arc Length
2
Converting between Degrees and Radians
Convert to Degrees
a)
b)
c)
3
4
7
5
a)
b)
c)
15
300
100
5
6
2. Convert to Radians
Radians and Arc Length
Arc Length (Degrees)
π
3
5 cm
Arc Length (Radians)
s
π
2
16 cm
s
Radians & Arc Length
3
What is the length of an arc if the circle
What the length of the an arc if the circle
has a radius of 4 cm and the central angle has a radius of 10 cm and the central
4
is 120?
angle is
?
5
Find the measure (in radians) of a central angle that intercepts an arc of length 4 cm
in a circle of radius 10 cm.
Application Problems
PULLEY -- A weight is lifted by a pulley as it rotates. As it rotates the rope holding the weight
gets wrapped along the circumference of the pulley thus causing it to lift of the weight.
Example 1 -- If the radius of the pulley is 10 cm and the rotation
angle of the pulley was  radians, how high will the weight rise?
4
GEARS – Two gears are locked together by their cogs or teeth. When one gear moves the
other moves as well. While the two gears of different size rotate different amounts due to
their size, what they do have in common is the amount of arc length that is used while they
turn.
Example 2 – There are two gears one with a radius of 6 cm and the
other with a radius of 24 cm. If the angle rotation of the smaller gear
was 2 radians, what was the angle of rotation in radians of the larger
gear?
Radians & Arc Length
4
HOMEWORK
1. Convert the degree measures into radians. Leave answers as exact values in most reduced form.
a) 90
b) 30
c) 300
d) 270
__________ radians
__________ radians
__________ radians
__________ radians
2. Convert the following radian measures into degrees.
a)
5
3
b)
__________  
e)

c)
__________  
6
5
f)
__________  
9
20

11
12

g)
__________  
4
15
__________  
3
10
d)

h)

__________  

7
6
__________ 
4
2
__________
3. Determine the arc length.
a) Central Angle of 30,
radius of 3 cm
b) Central Angle of 90,
radius of 8 cm
c) Central Angle of 72,
radius of 10 cm
s = ____________ (E)
s = ____________ (E)
s = ____________ (E)
d) Central Angle of

4
rad . ,
e) Central Angle of
2
rad . ,
3
f) Central Angle of
4
rad . ,
5
radius of 12 cm
radius of 15 cm
radius of 10 cm
s = ____________ (E)
s = ____________ (E)
s = ____________ (E)
Radians & Arc Length
5
4. Determine the arc length of the following.
a)
b)
4π rad.
5
10 cm
s = ____________ (E)
c)
7π rad.
6
3 cm
s = ____________ (E)
d)
3π rad.
2
π rad.
6
4 cm
s = ____________ (E)
18 cm
s = ____________ (E)
5. Find the measure (in radians) of a central angle that intercepts an arc of length 16 cm
in a circle of radius 8 cm.
6. Find the measure (to the nearest degree) of a central angle that intercepts an arc of length 2.4 cm
in a circle of radius 10 cm.
7. If the radius of the pulley is 12 cm and the rotation of the pulley was
7
6
radians, how many cm will the weight rise?
8. If the arc length of the pulley was 8 cm and the rotation of the pulley
4
was
radians, what was the radius of the pulley?
5
Radians & Arc Length
9. The rotation of the smaller gear with a radius of 10 cm was
6
11
radians. What was the angle of
6
rotation (radians) of the larger gear with a radius 20 cm?
10. Find the radius of the larger gear in the figure if the smaller gear rotates 120 and the larger gear
rotates 30. Show all work.
Radius Small Gear = 5 cm
11. Find the angle of rotation of the smaller gear (in radians) if the larger gear rotates 180. Show all work
Radius Small Gear = 5 cm
Radius Large Gear = 12 cm
Did the small gear rotate more than one full revolution? Yes or No