Chapter 6
Section 6.1
Radian Measure
The Unit Circle
The unit circle is a circle of radius 1 with it center at the
origin. The equation of the unit circle is x2+y2=1. Any point
on the unit circle will have the sum of the squares of its x
and y coordinates equal to 1.
Terminal Points and Radian Measure
A terminal point on the unit circle is a point on the unit
circle that forms an angle with the positive x-axis. The
distance you travel on the unit circle starting from the point
(1,0) on the positive x-axis and ending at the terminal point
(x0,y0) is the radian measure of the angle. (Remember the
measure is positive if you move counterclockwise and
negative if you move clockwise.) The radian angle measure
we usually denote with the letter t.
y
x2  y 2  1
1
x
 t
x0 , y0 

1
The length of the red arc
above is the radian measure
of the angle in standard
position with the point
(x0,y0) on it terminal side.
The symbol  used in radian measure stands for the number   3.1415926…. This number is
irrational (i.e. its decimal expansion will never end or repeat). The reason that radian measure is
used more often in mathematics, physics, engineering and other disciplines is that the angle
measure is the length of arc (part of a circle) of a unit circle ( a circle of radius 1). Radian
measure represents a physical distance.
y
x
1
1/8 circle
1/8 · 2 = /4

r
The part of the unit circle marked in red is called an arc. The length of this
(if you straighten it out) is /4 or the measure of the angle in radians. This
can be related to a number by the following calculation:

4
   4  0.7853981633...
For an arc that is part of a circle of radius r its length often we use the
symbol (s) can be found by taking s=·r where  is the measure of the
angle in radians.
2𝜋
2𝜋
24 =
∙𝑟
An arc of radian measure is of length 24.
3
3
36
What is the radius of the corresponding circle?
𝑟=
≈ 11.4591
𝜋
Radian Measure
The radian
measurement of
an angle is based
on the radian
measure of an
entire circle being
2 and any
fraction of the
circle will be
proportional (i.e.
will form the same
fraction). The
examples below
show the measure
of angles in
standard position.
y
y
x
y
x
y
x
x
½ circle
¼ circle
1/8 circle
5/8 circle
½ · 2 = 
¼ · 2 = /2
1/8 · 2 =
/4
5/8 · 2 =
5/4
y
y
x
y
x
y
x
x
1 circle (neg)
3/8 circle (neg)
1/12 circle (neg)
¾ circle (neg)
1 · -2 = -2
3/8 · -2 = -3/4
1/12 · -2 = -/6
¾ · -2 = -3/2
t

2
t 
1
3
2
t
1
t
1
t

3
4
4
1
1
t
1
1

2
t
 5
4
The pictures above illustrate different angles on the unit circle along with their radian
measure which is also the length of the red arc. What are the measures of the last 4 angles?
Points and Angles
t
There are some angles on the unit circle for which we know
the coordinates of the terminal point. These come from
realizing the triangle formed by the terminal point the point
perpendicular on the x-axis and the origin is either a 30°60°-90° triangle or a 45°-45°-90° triangle. In radians we
would say they are:
  
/
/
6 3 2
or

2
; (0,1)
t
1 3

;  ,
3  2 2 

t
 2 2

; 
,
4  2 2 

t
 3 1
; 
, 
6  2 2 

  
/
t  0; (1,0)
/
4 4 2
0
1
Converting Angle Measure
Both radians and degrees are based on a fraction of a circle that you are considering.
Fraction of circle
1/12
1/8
1/6
1/4
Degree Measure
30
45 60
90
120 135 150 180 270
Radian Measure
/6
/4
/2
2/3
90 and /2
120 and 2/3
60 and /3
45 and /4
135 and 3/4
30 and /6
150 and 5/6
180 and 
0 and 0
/3
3/8
5/12
3/4



360
2
360  
 
2
2  

360
1/2

5/6
To convert back and forth
from degrees to radians we
use the proportion below
where  is the measure in
degrees and  is the
measure in radians.

270 and
3/2
1/3

3/4
3/2
2  30 2 1 


3600
12
6
2 120 2 1 2


360
3
3
360 

3  120  60
2
2
5
6  60  5  30  5  150
2
2
360 