Radian Measure of Angle

Angles are typically measured either in degrees or radians. In both cases, they are measures of portions of a circle of unspecified radius.

If you think of a circle in terms of a pie, and you slice the pie into 360 pieces, then a degree is the wedge opening of one slice of that pie. For radian measure of an angle, we specify the angle as a ratio of pie crust to pie radius (or arclength subtended of the circle's circumference to radius of the circle). If "s" is the arclength subtended (blue in the image to the left, click to enlarge), and "r" is the radius (red in the image), then the angle theta (yellow in the image), in radians, is s divided by r.In the animation shown below, you can select different angles and see what their values are in degrees and radians. Choose the radius by dragging on the right (or lower) black dot, and choose the angle by dragging on the left (or higher) black dot. "Theta sub d" is the angle in degrees, and "theta sub r" is the angle in radians. "Theta sub r times r" is s, the arclength subtended.