If we have the same central angle in two different circles, the length of the arc defined by the angle depends on the size of the circle. So, we can use the relationship between the arc length and the circle’s radius to get some information about the size of the central angle.

For example, suppose Circle A has radius 9 units and a central angle that defines an arc with length . Circle B has radius 15 units and a central angle that defines an arc with length . How do the two angles compare?

For the angle in Circle A, the ratio of the arc length to the radius is , which can be rewritten as . For the angle in Circle B, the ratio of the arc length to radius is , which can also be written as . That seems to indicate that the angles are the same size. Let’s check.

Circle A’s circumference is units. The arc length is of , so the angle measurement must be of 360 degrees, or 60 degrees. Circle B’s circumference is units. The arc length is of , so this angle also measures of 360 degrees, or 60 degrees. The two angles are indeed congruent.