Graphing
Calculator
Math
Worksheets
Algebara
Solver
Chart
Maker
A+ A− B

Convert Radians to Degrees

Formula to convert radians to degrees and back

What is the deal with radians anyway?

Answer: Most of you are used to thinking of a circle in terms of degrees: 360° is the whole circle. 180° is half the circle etc... Well, radian measure is just a different way of talking about the circle. Radian measure is just different unit of measure.

Just as we can measure a football field in yards or feet--we can measure a circle in degrees (like the good old days) or in radians (welcome to the big leagues!)

Think about what the word radian sounds like...well, it sounds like 'radius', right? It turns out that a radian has a close relationship to the radius of a circle


So what is a radian then?

Definition of radian : a radian is the measure of an angle that, when drawn a central angle of a circle, intercepts an arc whose length is equal to the length of the radius of the circle.

Demonstration of Radian of Circle

The video below is an animation of 1 radian. Notice how the length of 1 radius stretches out to a portion of the circle. That portion is 1 radian of the circle.



There is a simple formula to convert radians to degrees. 1Π radian = 180. Therefore you can easily convert from one unit of measure to the other

Degrees to radians

The general formula for converting from degrees to radians is to simply multiply the number of degree by Π /180°
Example 1:

Convert 200° into radian measure:
$ 200 ^{\circ} \cdot \frac{\pi }{ 180 ^{\circ}} \\ = \frac{10\pi}{ 9} \text{ radians } = \text{3.49 radians } $
Example 2:

Convert 120° into radian measure:
$ 120 ^{\circ} \cdot \frac{\pi }{ 180} = \frac{2\pi}{ 3}\text{ radians }= \text{2.09 radians } $

Radians to degrees

The general formula for converting from radians to degrees to simply multiply the number of degree by 180°/(Π)
Example 1:

Convert $$ \frac{4}{9} \pi \text{ radians} $$ to degrees $ \frac{4\pi}{9} \cdot \frac{180 ^{\circ} }{\pi} \\ = \frac{4\pi \cdot 180 ^{\circ} \cdot}{9\pi} =\frac{720 ^{\circ} \pi \cdot}{9\pi} \\ \frac{720 ^{\circ} \cancel{ \color{Red} \pi} \cdot}{9\cancel{ \color{Red} \pi}} \\ = 80 ^{\circ} $
Example 2

Convert 1.4 radians into degrees: $ 1.4 \cdot \frac{180 ^{\circ} }{\pi} \\ =\frac{252 ^{\circ} }{\pi} \approx 80.2^{\circ} $

So what's the deal with '$$\pi $$ radians' vs 'radians' ?

Answer: Let's rephrase the question as follows :
" Is there any difference between 5 radians and $$ 5\pi \text{ radians }$$? "
Answer

Practice Converting Degrees between Radians




Problem 1) What is the radian measure of 60°?
Answer



Problem 2) What is the degree measure of an arc whose measure is $$ \frac{2\pi}{3} \text{ radians} $$?
Answer



Problem 3) What is the degree measure of an arc whose measure is $$ \frac{7\pi}{12} \text{ radians} $$?
Answer


Problem 4) What is the degree measure of an angle whose measure is 14 radians?
Answer