Angle Measure in Radians


The ratio of the circumference of a circle to the diameter, or twice the radius, is denoted with the greek letter (pi).   is an irrational number, so we can only approximate it. The most common approximation is 3.14. Using modern methods, can be approximated to any number of decimal places.  

By definition, the circumference of a circle is , where is the radius of the circle. Since the radius of the unit circle is 1, the circumference of the unit circle is . Now suppose we trace an arc with a length of 1 on the unit circle. There will be a corresponding angle subtended by the arc. Since the length of the arc is one radius, we call the angle 1 radian. An angle is said to be in standard position if its apex is at the origin and the angle originates on the axis. A positive angle has counterclockwise rotation about the origin and a negative angle has clockwise rotation. The interactive figure below shows an arc of length 1, which subtends an angle of 1 radian. You can change the angle by moving the slider or typing the angle into the input box. Try and predict the arc and the angle, then type each of the following into the box and compare with your prediction. Resolve any differences before moving on.

Unit Circle Angles in Radians

Larger Angles

Until now, we have limited our discussion to angles between . Note that when , the angle appears to return to zero. The arc corresponding to the angle terminates at a point , called the terminal point on the unit circle. For angles greater than , we start another trip around the unit circle, resulting in angles that are co-terminal, meaning they have the same terminal points. Add to all of the angles above and note that the two angles are co-terminal. For example, are co-terminal because . Test this and other angles in the interactive figure below.