The Beginnings of
Chapter 4
Section 4.1a: Angles,
their measures,and
arc length
First, Some Definitions
Degree – represented by the symbol , is a unit of
angular measure equal to 1/180th of a straight angle.
Note: Each degree is subdivided into 60 minutes
(denoted by ’ ), and each minute is subdivided into
60 seconds (denoted by ” ).
This is called the DMS form for angle measure.
Conversion Problems
Convert each of the following from DMS to decimal form.
422436
Each minute is 1/60th of a degree, and each second is 1/3600th
of a degree:
24   36 

422436  42      
   42.41
 60   3600 
60556
5   56 

60556  60      
   60.099
 60   3600 
Conversion Problems
Convert the following from decimal to DMS form.
37.425
We need to convert the fractional part to minutes and seconds.
First, convert the 0.425 degrees to minutes:
 60 
0.425 
  25.5
 1 
Then, convert 0.5 minutes to seconds:
60 

 30
0.5 

 1 
Final Answer:
37.425  372530
Radians
So, what’s the problem with degrees, anyway???
 Degree units have no mathematical relationship
whatsoever to linear units!!!
EX: There are 360 degrees in a circle with radius 1…
What relationship does the 360 have to the 1???
Is it 360 times as big???
Enter RADIANS to help solve these dilemmas…
Definition: Radians
A central angle of a circle has measure 1 radian
if it intercepts an arc with the same length as the
radius.
a
1 radian
a
Degree-Radian Conversion
To convert radians to degrees, use the conversion
factor: 180   radians
Note:
π radians and 180
both measure a straight angle!!!
We will use dimensional analysis to convert all angles.
Guided Practice
How many radians are in 90 degrees?
  rad  90
90 

 180  180
How many degrees are in 

radians 
2
radians
3 radians?

  180  180
 60
 rad  

3
3
   rad 
More Definitions
In navigation, the course or bearing of an object is
sometimes given as the angle of the line of travel
measured clockwise from due north.
Ex: Sketch a diagram of the path of a boat leaving a harbor
with a bearing of 155
Harbor
155
Path
of boat
ARC LENGTH FORMULA (RADIAN MEASURE)
If 0 is a central angle in a circle of radius r,
and if 0 is measured in radians, then the length
s of the intercepted arc is given by
s  r
s  arc length (linear unit of measure)
 = measure of angle of rotation in radians
r = radius (linear unit of measure)
ARC LENGTH FORMULA (DEGREE MEASURE)
If 0 is a central angle in a circle of radius r,
and if 0 is measured in degrees, then the length
s of the intercepted arc is given by
s
 r
180
GUIDED PRACTICE
Use the appropriate arc length formula to find the
missing information.
s
r
70 cm
1 cm

70 rad
2.5 ft 7.5  ft  3 rad
4 in
7 in
4 7 rad
 2m
5m
18
GUIDED PRACTICE
Find the perimeter of a 60 slice of a large (7 in. radius)
pizza.
Delicious slice of pizza: Perimeter:
7 in + 7 in + s in
Find s :
s in
7 in
s
60
7 in
  7  60 
180
7

 7.330
3
The perimeter is approximately 21.330 inches
GUIDED PRACTICE
The running lanes at a certain track are 1 meter wide. The
inside radius of lane 1 is 33 meters and the inside radius of
lane 2 is 34 meters. How much longer is lane 2 than lane
1 around one turn?
Lane 2
Lane 1
33 m
34 m
Each lane is a semicircle with
central angle
and
length
.
 
s  r  r
Therefore, the difference in
their lengths is
34  33  
Lane 2 is about 3.142 meters longer than lane 1.
GUIDED PRACTICE
It takes ten identical pieces to form a circular track for a
pair of toy racing cars. If the inside arc of each piece is
3.4 inches shorter than the outside arc, what is the width
of the track?
What is the measure of each of
these central angles?
  5

Inside arc length for one piece
of track:
ri  5
Outside arc length for one piece
of track:
o
r  5 
GUIDED PRACTICE
It takes ten identical pieces to form a circular track for a
pair of toy racing cars. If the inside arc of each piece is
3.4 inches shorter than the outside arc, what is the width
of the track?
ri  5
ro  5
But we were given the difference
between these arc lengths:

ro  5  ri  5  3.4
ro  ri  3.4  5  
 5.411 inches
GUIDED PRACTICE
The concentric circles on an archery target are 6 inches
apart. The inner circle (red) has perimeter of 37.7 inches.
What is the perimeter of the next-largest (yellow) circle?
Perimeter of inner circle:
 d  37.7
Perimeter of outer circle:
6
d
6
  d  6  6   d  12
 37.7  12
 75.399 inches
Whiteboard Conversion Problems
Convert each of the following from DMS to decimal form.
343018
30   18 

343018  34      
   34.505
 60   3600 
1191537
15   37 

1191537  119      
  119.260
 60   3600 
Whiteboard Conversion Problems
Convert each of the following from decimal to DMS form.
10.98  105848
Convert the 0.98 degrees
to minutes:
 60 
0.98 
  58.8
 1 
Convert the 0.8 minutes
to seconds:
60 

0.8 
 48

 1 
25.29  251724
Convert the 0.29 degrees
to minutes:
 60 
0.29 
  17.4
 1 
Convert the 0.4 minutes
to seconds:
60 

0.4 
 24

 1 
Whiteboard Practice
Convert each of the following from DMS to radians.

5
(a) 150

rad
180 6
(b)
7530   75  30 60   75.5
75.5

180
 1.318rad
Whiteboard Practice
Convert each of the following from radians to degrees.
7

180

(a)
 126
10 
(b) 1.3
180

 74.485
WHITEBOARD PRACTICE
Convert from DMS to radians
11.83

180
 0.206 rad
Convert from radians to degrees
13 180
 117
20 