One way to define a circle that is centered at is by the equation , where is the radius of the circle. A unit circle has , so the equation for this unit circle with center must be . Points on the unit circle have several interesting properties, such as having matching points on opposite sides of the axes due to symmetry. Another feature of points on a unit circle is that they can be defined solely by an angle of rotation that is measured in radians.

Radians are a natural tool to use to measure the distance traveled on a circle. Let’s say that the wheels on a bike have a radius of 1 foot. When the bike starts to move to the left, rotating the wheel counterclockwise, let’s think about what happens to point .

The point will return to its starting location when the wheel has rotated through an angle of  radians. During this rotation, the bike will move a length equal to the circumference of the wheel, which is feet. In general, the angle of rotation of the wheel with a radius of 1 foot, in radians, is the same as the number of feet this bike has traveled. So what do we do when a wheel doesn’t have a radius of 1 unit? Since all circles are similar, we can use the same type of thinking—scaled up or down—to match the size of the wheel, which is something we’ll do in future lessons.

Thinking about the wheel as a unit circle, as shown in this image, the arc length of the circle from to has length equal to 1 unit, the radius of the unit circle. Because of this, the angle is said to measure one radian. If we continue to measure off radian lengths around the circle, it takes a little more than 6 to measure the entire circumference.

This makes sense because the ratio of the circumference to the diameter for a circle is , and so the circumference is times the radius, or about 6.3 radii.

Let’s think about some other angles on the unit circle. Here, angle measures radians because its arc is of a full circle (counterclockwise) or of . Angle is three quarters of a full circle (counterclockwise), so that’s radians.