Radians and Degrees

by: Richard Forbes, Kara Kenney

Professer Moosifur

Radians and Degrees song lyrics

(just in case you couldn't make them out,
read them here with all their genius exposed)

3.14 (dot dot dot) is the number
Of Radians in a semicircular
In a full circle, radians double
Pi turns into 2 pi just to make trouble

Radians radians used by Canadians

Degrees degrees make me sneeze (achoo)

Degrees divide a circle up odd

360 degrees in a circle made of cod
Divide that in two, and you find a semicircle
180 degrees that encircles

Radians radians – there everythings arcadian
Degrees degrees they are quite a breeze (woosh)

Switching back and forth from one to the other
Will quickly make you dread another
Calculators make it easy
But then math teachers couldn’t keep us busy

Radians radians- nothing to do with circadian
Degrees degrees they are the bee’s knees (bzzz)

To get from radians to degrees you use 180 over pi
if there are a lot you will want to sigh
For radians to degrees, you use the reciprocal
Doing this a lot is also very dull.

Radians radians used by Canadians
Degrees degrees make me sneeze (achoo)

There are two different ways of measuring angles. One uses degrees, and one uses radians. Both methods are described here.


Radians are a process of measuring angles. Instead of degrees, one uses the length of the radius. The angle that radius covers has been found to be a little more than 3. π radians make up a semicircle. All angles are based off this information.
360 degrees = 2 π or 0 π
180 degrees = π
90 degrees = π/2
45 degrees = π/4
30 degrees =π/6


Degrees don't use a ratio, but are instead a fraction of the circle. One degree is 1/360 of a revolution around the vertex. A full revolution is 360°, half of a revolution is 180°, and a quarter of a revolution is 90°.

The simplest way to convert radians to degrees is by multiplying the radian measure by 180/π. The simplest way to convert degrees to radians is by multiplying the degree measure by π/180.

Finding Arc Length, Linear Speed, and Angular Speed:

An angle can also be used to find the length of an arc. When using radians, the formula s = rӨ can be used to find the length of the arc. "s" is the length of the arc, "r" is the radius of the circle, and Ө is the angle in radians. This formula will not work with an angle in degrees, so the angle should be changed into degrees before this forumla is used.

The arc length can also be found by using the formula s = 2πr.

After the arc length has been found, it can be used to find linear and angular speed.
Linear speed can be found with the formula: arc length/time, where "arc length" is the length of the arc that is passed through.
Angular speed can be found with the formula: central angle/time, where "central angle" is the angle that is passed through.


Answers: 45° & -315°, π/4 & -7π/4

Answers: 210° & -150°, 7π/6 & -5π/6

Change the following radian angles into degrees:
a) 3π/2
b) 7π/3
c) π/5
d) 5π/6

a) 270°
b) 420°
c) 36°
d) 150°

Change the following degree angles into radians:
a) 30°
b) 90°
c) 145°
d) 240°

a) π/6
b) π/2
c) 29π/36
d) 4π/3

Find the complement and supplement of the following angles:
a) π/3
b) π/12
c) 36°
d) 23°

a) π/6, 2π/3
b) 5π/12, 11π/12
c) 54°, 144°
d) 67°, 157°

The second hand of a clock is 11.4 centimeters long. (Remember - a second hand takes 60 seconds to complete one revolution)
How far does the tip of the hand travel in one minute?
At what speed does the tip travel?
What is the angular speed of the tip?

The tip travels 71.6 centimeters, or 22.8π centimeters.
It travels at the rate of 1.19 centimeters per second.
The angular speed is π/30 radians per second, or 6° per second.