There are three units of measure for angles: revolutions, degrees, and radians. In trigonometry, radians are used most often, but it is important to be able to convert between any of the three units.

A revolution is the measure of an angle formed when the initial side rotates all the way around its vertex until it reaches its initial position. Thus, the terminal side is in the same exact position as the initial side. In trigonometry, angles can have a measure of many revolutions--there is no limit to the magnitude of a given angle. A revolution can be abbreviated "rev".

A more common way to measure angles is in degrees. There are 360 degrees in one
revolution. Degrees can be subdivided, too. One degree is equal to 60
minutes, and one minute is equal to 60 seconds. Therefore, an angle
whose measure is one second has a measure of
degrees. When
perpendicularity is discussed, it is most
often defined as a situation in which a 90 degree angle exists. Often degrees
are used to describe certain triangles, like
30-60-90 and
45-45-90 triangles. As previously mentioned,
however, in most cases that concern trigonometry, radians are the most useful
and manageable unit of measure. Degrees are symbolized with a small superscript
circle after the number (measure). 360 degrees is symbolized
360^{
o
}
.

A radian is not a unit of measure that is arbitrarily defined, like a degree.
Its definition is geometrical. One radian (1 rad) is the measure of the
central angle (an angle whose vertex is the center
of a circle) that intercepts an arc whose length is
equal to the radius of the circle. The measure of such an angle is always the
same, regardless of the radius of the circle. It is a naturally occurring unit
of measure, just like
*Π*
is the natural ratio of the circumference of a
circle and the diameter. If an angle of one radian intercepts an arc of length
*r*
, then a central angle of
2*Π*
radians would intercept an arc of length
2*Πr*
, which is the circumference of the circle. Such a central angle has a
measure of one revolution. Therefore,
1 rev = 360^{
o
} = 2*Π* rad
.
Also,
1 rad = ()^{
o
} =
rev.

Below is a chart with angle measures of common angles in revolutions, degrees, and radians. Any angle can be converted from one set of units to another using the definition of the units, but it will save time to memorize a few simple conversions. It is particularly important to be able to convert between degrees and radians.

Figure %: Some common angles measured in all three units of measurement