Definition: A radian is the angle at the center of a circle that cuts off an arc exactly as long as the radius; a full turn measures 2 pi radians.
What is a radian?
Take a circle of radius r and mark an arc of length r. The central angle that opens to that arc is 1 radian, about 57.3 degrees. Since the circumference is 2 pi r, a full turn is 2 pi radians, and a half turn is pi.
How do you convert?
- Degrees to radians: multiply by pi/180.
- Radians to degrees: multiply by 180/pi.
Worked example: convert 105 degrees. 105 times pi/180 = 7 pi/12 radians, about 1.833.
Radians also make arc length simple: s = r times theta. A circle of radius 9 cm and angle 7 pi/12 gives s = 9 times 7 pi/12 = 21 pi/4, about 16.5 cm.
What goes wrong?
Leaving a calculator in degree mode when the problem uses radians. Mathbot checks the mode first, since sin(2) gives very different answers in the two settings.