In everyone's experience it is usual to measure angles in degrees. We learn early in childhood that there are 360 degrees in a circle, that there are 90 degrees in a right angle, and that the angle of an equilateral triangle contains 60 degrees. On the other hand, to scientists, engineers, and mathematicians it is usual to measure angles in radians.

The size of a radian is determined by the requirement that there are 2 radians in a circle. Thus 2 radians equals 360 degrees. This means that 1 radian = 180/ degrees, and 1 degree = /180 radians.

The reason for this is that so many formulas become much easier to write and to understand when radians are used to measure angles. A very good example is provided by the formula for the length of a circular arc. If A and B are two points on a circle of radius R and center C, then the length of the arc of the circle connecting them is given by

d(A,B) = R a,

where R is the radius of the sphere, and a is the angle ACB measured in radians. If we measure the angle in degrees, then the formula is

d(A,B) = R a/180,

These formulas can be checked by noticing that the arc length is proportional to the angle, and then checking the formula for the full circle, i.e., when a = 2 radians (or 360 degrees).

The figure below gives the relationship between degrees and radians for the most common angles in the unit circle measured in the counterclockwise direction from the point to the right of the vertex. The form of the ordered pair is {degree measure, radian measure}

Created on The Geometer's Sketchpad by Boyd E. Hemphill