Radian Measure

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 clockwise direction). 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.



Coterminal Angles Revisited

We have already defined coterminal angles in terms of revolutions and what that means in degree measure. For angles in radians, two angles are coterminal if they differ by a multiple of 2. We can list all angles coterminal to the angle x using the shorthand notation x + 2n, n an integer. Of course, we can also list the angles using the longer form, namely:
... , x - 6, x - 4, x - 2, x, x + 2, x + 4, x + 6, ...





Numerical approximations



Sometimes it is confusing to see an angle measured in radians shown as a number involving and as a decimal approximation. For example, the angle of size /4 is also the angle of size (approximately) 0.7854. In this exercise, you are asked to find a decimal approximation to an angle written in terms of .

The next two exercises are for you to practice converting angles from radians to degrees and vice-versa.


Radians to Degrees



Here you are given an angle in radian measure and you are to convert it to degree measure. All the angles have been chosen so that the degree measurement is an integer. Your answer should be written with the degree symbol o. If you are not sure how to type that in, put your mouse over the "?" symbol.



Degrees to Radians



This exercise requires you carry out the reverse steps of the previous exercise. That is, you are given an angle measured in degrees and you are requested to write it as an angle in radians. The answer is to be EXACT and not a decimal approximation. Therefore, your answer needs to be a rational multiple of