The radian measure of an angle drawn in standard
position in the plane is
equal to the length of arc on the unit circle subtended by that angle.
In the figure, the angle drawn subtends an arc length of size t on the unit circle so the radian measure of the angle is also t.
Recall that the symbol represents the real number constant which is the ratio of the circumference of a circle to its diameter. Its value is approximately 3.14159. Since the circumference of the unit circle is 2 it follows that the radian measure of an angle of one revolution is 2. The radian measure of an angle whose terminal side is along the negative x -axis is .
We now can easily obtain a formula to convert from degrees to radians and vice-versa.
|Conversion between degrees and radians|
To convert from degrees to radians, multiply the angle by /180.
To convert from radians to degrees, multiply the angle by 180/.
|Angles in degrees and radians|
In this example, click on or within the unit circle. You will be shown the
angle you selected measured in both degrees and radians. Also you are given the
negative angle (had the rotation of the terminal side been done in the
Once you have clicked once to get an angle, you may drag the angle rather than
click again. The angle shown is always less than one revolution.
The next two exercises are for you to practice converting angles from radians
to degrees and vice-versa.
|Radians to Degrees|
|Degrees to Radians|