In most of this course we are discussing the motion of a small object, and we describe this in terms of the object's position, which we might measure in meters. However, many everyday objects are rigid and have an unchanging shape; now we can discuss the orientation of the object, measured as an angle.

There are several ways to measure angles. For example, we can relate it to a whole circle: someone has eaten ¼ of this pie.

The drawback to this approach is that we usually are interested in small fractions of a circle. People are not good at thinking in fractions! The solution is to use a small standard fraction of a circle, such as the degree (which is 1/360 of a whole circle). The choice of 360 as the divisor was made by the Babylonians and is somewhat arbitrary, but it has the advantages that when we divide a pie into 2, 3, 4, 5, 6, 8, 9, or 10 wedges we get pieces with an integer number of degrees at the center vertex. It is also useful that the degree is a small unit. In everyday uses we rarely need angles smaller than 1 degree: when we say "the tabletop is level" we usually mean that its slope is at most a degree or two. But the full moon is only ½ degree wide, and some people can see objects that are only 1/10 degree wide.

As a wagon moves forward, the wheels turn. Since the circumference of the wheel is 2 pi r = 6.28 r, that is how far the wagon advances per turn of the wheel. The numerical factor can be hidden if we measure angles in radians, which is an angular measure defined so that 1 whole circle = 2 pi radians. Then the relationship between angle and distance is just
distance cart moves = (radius of the wheel) x (angle the wheel moves, measured in radians)