Algebra II (p.26)
F.TF.1 Understand radian measure of an angle as the length of the arc on the
unit circle subtended by the angle.
Intro:
Welcome. In this session, we will focus on developing a conceptual understanding of
radian measure, and develop a method for converting between degrees and radians.
The ideas referenced in this session were inspired by the following article:
Kalid Azad’s Intuitive Guide to Angles, Degrees, and Radians, which can be found at
http://betterexplained.com.
Concept Development:
When we measure quantities (such as length, weight or volume), we must first
determine what our unit of measure will be. Ask students what unit of measure
they would use to measure the length of a rectangular swimming pool.
Students may suggest measuring the pool length in feet or meters, but encourage
examples of units that seem somewhat arbitrary as well. For example, if we didn’t
have a tape measure with us at the pool, we could take off our flip-flop and measure
the length of the pool in terms of flip-flops.
Later, I could measure my flip-flop in terms of some other unit (like inches) and
convert the length of the pool in terms of the new unit. Any fixed length (like my
beach towel or my sunblock, for instance) could be used as a unit of measure, and
then converted in terms of any other unit of measure.
Suppose a coach is standing in the middle of a circular track watching an athlete
running around the track from point A to point B. How do we measure the angle
formed by the runner’s path?
Students are used to measuring angles in terms of degrees, and we can think of this
as the amount of rotation between two points on a circle, where the center of the
circle is the point of rotation. In this case, we’re measuring the angle from the
coach’s perspective. The coach would measure the angle by standing in the center of
the circle and rotating her head from point A to point B.
Let’s measure the angle from the runner’s perspective. The runner starts at point A
and runs a certain distance until he reaches point B. From his perspective, the angle
is determined by the distance he traveled.
For a given circle, there is a one-to-one correspondence between any given angle
measure and the length of the arc that subtends the angle (in other words, if two
angles in a given circle have the same measure, then the arcs formed by those angles
must have the same length). Therefore, the runner’s method for measuring the angle
is just as valid as the coach’s method.
However, the runner wouldn’t give the absolute distance he traveled, because in
that case, different-sized tracks would produce different-sized angles. For instance,
traveling a quarter mile on a quarter-mile track would mean he completed exactly
one lap, which would correspond to an angle of 360 degrees, but traveling a quarter
mile on a half-mile track would mean he completed exactly half a lap, which would
correspond to an angle of 180 degrees. So his absolute distance would not
necessarily determine the angle measure. 
Instead, the runner would give the distance he traveled relative to the size of the
circle, which is determined by the length of the circle’s radius.
If we let θ represent the angle measure, s represent distance traveled along the
circle (or arc length), and r represent the radius of the circle, then
s
r
 .
For example, if the runner is running on a quarter-mile circular track (440 yards),
then the radius of the track is approximately 70 yards. If the runner runs 100 yards,
what angle is formed by the starting and finishing lines?

100 yards
1.43
 1.43 
70 yards
1
In simplifying the ratio, the units cancel out. This will happen as long as we use the
same unit for measuring radius as we do for measuring circumference (and in turn,
arc length). This gives an angle of approximately 1.43. This is not 1.43 degrees, but
rather 1.43 radians.
How big is 1.43 radians? In a unit circle, it’s the angle that corresponds to an arc of
length 1.43.
How big is one radian? In order for an angle to equal exactly 1 radian, what must be
true about s and r? s and r must be the same length. A radian, then, is the size of an
angle such that the distance traveled is equal to the length of the radius.
To reiterate, degrees measure angles in terms of rotation, whereas radians measure
angles in terms of distance traveled.
When we measure the size of an angle in radians, we use the radius as our unit of
measure. Just like we can measure the length of a swimming pool in terms of flipflops, we can measure the length of an arc in terms of a radius. And just like we can
convert flip-flops into inches, we can convert degrees into radians.
Let’s make a table showing the relationship between degrees and radians for several
angle measures.
Degree measure
Radian measure:  
360°
2  6.28
180°
  3.14
90°
60°
45°
30°
15°
1°

2

3

4

6
 1.57
 1.05
 0.79
 0.52

12
 0.26

180
 0.02
s
r
Now that we know 1° is equivalent to

180
radians, we can easily convert any
degree measure to radians. For example, 36  36  1  36 
multiplying the degree measure by

180
180

180


5
radians . So
gives the corresponding radian measure.
Likewise, dividing the radian measure by
by


180
(which is equivalent to multiplying
) gives the corresponding degree measure.
So how big is one radian? 2  360 , so dividing both sides by 2 gives
180
1
 57.3 . Visually, we can estimate the size of one radian by wrapping a

radius length around the circle.
1.43  1.43  1  1.43 
180

 81.9
Developing the method for converting between degrees and radians in this way

rather than simply providing the rule (multiply degrees by
) should help
180
students deepen their understanding of the relationship between the two
representations. Often times students are confused by angles expressed in radian
measure because they don’t understand the meaning of a radian, or they don’t know
why they should use radians instead of degrees.
So why use radians? Radians are often more convenient in problems involving
angular motion, because such problems often involve distance traveled. Say, for
example, you’re riding on a bus with wheels of radius 2 feet. The bus is traveling 12
feet per second. How fast are the wheels turning? In one second, the distance
traveled is 12 feet, so if we divide that by the radius of 2 feet, we get the angle of
rotation, 6 radians. So the wheels are turning at an angular speed of 6 radians per
second. Our calculations would have been much more complicated if we had been
working in terms of degree measure.
Teaching Implications:
In summary, we have examined a method for measuring angles in terms of distance
traveled along the edge of a circle, and have devised a method for converting
between radian and degree measure. This wraps up our session. Thank you for
joining us.