Units and Angles

The best way to measure angles is with "units" called radians. The quotation marks around the word "units" are very important. We measure an angle like the angle of the blue sector in the picture at the left by dividing the length, L, of the arc cut-off by the angle by the length, R, of the radius of the circle. Since we are dividing two quantities, both of which are measured in the same units of length, the result is a dimensionless quantity -- that is, a quantity that has no associated units. This observation is important when we look at many familiar formulas involving angles. For example, the length of the arc of a circle whose radius is R that is cut-off by the angle theta is pi R theta Since R is measured in units of length and neither pi nor theta have associated units, the net result of this computation is measured in units of length.

The usual trigonometric functions -- sine, cosine, tangent, and so forth -- are also dimensionless since they are quotients in which both the numerator and denominators are measured in units of length.