Radians & Degrees on the Unit Circle

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Angles are important in all kinds of fields, such as construction, science, math, navigation, and engineering. There are two units that we can use to measure angles: degrees and radians. In this lesson, learn how to use these units and how they are related to the unit circle.

Measuring Angles

Imagine that you are camping with some friends and need to follow precise directions to reach your campsite. Your friend said if you walk in a direction 43 degrees east of north from the parking lot, you will reach the campsite in about thirty minutes. You have a compass, but how do you know exactly where to go? You would need to be able to measure an angle of 43 degrees from north in order to follow these directions correctly. Being able to find and use angle measurements is important not only for helping you to find your campsite, but also in fields as diverse as construction, astronomy, and engineering.

There are two main unit systems that we use for measuring angles: degrees and radians. One degree is 1/360 of a complete circle, so there are 360 equally divided degrees in a circle. Why do we divide the circle into 360 degrees instead of some other number? We aren't really sure why this number was chosen originally, but we think the 360 degree circle was first used by ancient Babylonians over 3000 years ago. Some scholars think it was based on the length of a year (which is actually 365 days), but the true origins of the degree are lost to history.

Although degrees are still used in many applications, radians are often preferred to degrees in higher mathematics because they can be more easily used in a variety of calculations. The radian is a more modern angle measurement that is based on the arc length of a unit circle. Before you can really understand what that means, we need to spend a few minutes learning what a unit circle is.

Radians and The Unit Circle

The unit circle is a circle centered at the origin (0,0) that has a radius of exactly one. It is used to define the radian unit and to define the trigonometric functions. We won't go into the trigonometric functions in this lesson, but there are many more lessons on those if you want to check them out!

The unit circle has a radius of 1
The Unit Circle

To define the radian, it is also important to understand what a circular arc is and how its length is measured. An arc is created when two radial lines are drawn from the center of the circle at a certain angle apart from each other. The part of the circle inside these two lines is called a circular arc, or sometimes just an arc.

One radian is defined as an angle that would give an arc length of one on the unit circle.

Angles can be measured in units called radians, and the measure of an angle in radians is equal to the length of an arc (s) on the unit circle that subtended by the angle
relationship between angle and arc length

Converting Between Radians and Degrees

Angles can be measured in units of radians or degrees, and it is important to be able to use both and to convert between them.

If you went all the way around the circle, the arc length would be equal to the entire circumference of the circle.

c = 2pi x r = 2pi x 1 = 2pi

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