How to convert from degrees to radians
To convert from degrees to radians, just multiply by (π radians) / (180 degrees). The math is shown below.
Examples showing how to convert from degrees to radians
Example #1: Convert 30 degrees to radians
30 degrees = 30 degrees × π radians / 180 degrees
= (30 degrees / 1) × (π radians / 180 degrees)
= (30 degrees × π radians) / 1 × 180 degrees
= (30 π radians) / 180
= π radians / 6
Notice that since degrees is on top and at the bottom, we can cancel it. Notice also that we divided both the numerator and denominator of (30 π radians) / 180 by 30 to simplify it.
Example #2: Convert -90 degrees to radians
-90 degrees = -90 degrees × π radians / 180 degrees
= (-90 degrees / 1) × (π radians / 180 degrees)
= (-90 degrees × π radians) / 1 × 180 degrees
= (-90 π radians) / 180
= -π radians / 2
Example #3: Convert 120 degrees to radians
120 degrees = 120 degrees × π radians / 180 degrees
= (120 degrees / 1) × (π radians / 180 degrees)
= (120 degrees × π radians) / 1 × 180 degrees
= (120 π radians) / 180
= 2π radians / 3
Did you understand the examples above clearly? Did you not understand them at all? The figure below summarizes the concept.
Why did we multiply by (π radians) / (180 degrees) when converting from degrees to radians?
We know that 360 degrees = 2π radians
Just divide both sides of the equation 360 degrees = 2π radians by 2.
(360 degrees)/2 = (2π radians)/2
180 degrees = π radians
Now let ? represent the amount in degrees that you would like to convert to radians and x the converted amount in radians. You can set up first an "if-then" sentence.
"If" 180 degrees equal π radians,
"then" ? is equal to x .
Then, you can pull out all the numbers in red and put them in a table in the same order they are listed above.
| 180 degrees | π radians |
| ? | x |
The proportion is 180 degrees / ? = π radians / x
After cross multiplication, we get 180 degrees × x = ? × π radians
x = (? × π radians) / 180 degrees
x = ? × (π radians / 180 degrees)
Therefore, if ? is the amount in degrees, to convert from degrees to radians, just multiply by (π radians / 180 degrees)
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