This lesson covers two big topics:

1. What is an angle?

2. How do we measure angles?

1. What is an angle?

Angles are a dynamic concept. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex as shown in both pictures below. One is drawn in open space and the other is drawn on a coordinate plane in standard position (this is how we will roll in this class). An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. It is possible for more than one angle to have the same terminal side. In the third picture below, the angle of 140° is a positive angle, measured counterclockwise. The angle of –220° is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal. Every angle greater than 360° or less than 0° is coterminal with an angle between 0° and 360°, and it is often more convenient to find the coterminal angle within the range of 0° to 360° than to work with an angle that is outside that range.

Illustration of an angle with labels for initial side, terminal side, and vertex.

Graph of an angle in standard position with labels for the initial side and terminal side.

2. How do we measure angles?

Degree Measure

Radian Measure

We can measure an angle using degrees. Rotation from the initial side all the way around to the initial side again is 360 degrees. That is, one degree is 1/360 rotation around the vertex. The measure, in degrees, of an angle is how many degrees the angle rotates from the vertex (center of circle).

One radian is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360°) equals $2 π$ radians. A half revolution (180°) is equivalent to $π$ radians.

Related Degrees and Radians

\begin{array}{l} 1 rotation = 360° & = 2 π & radians \\ \frac{1}{2} rotation = 180° & = π & radians \\ \frac{1}{4} rotation = 90° & = \frac{π}{2} & radians \end{array}

Converting from Degress to Radians:

Since radians and 180° both measure the same angle, we can see that rad = 180°. This means that
1° = rad and 1 rad = {}°

This means that:

To convert from degrees to radians, we multiply by

For example, 60° is converted to /3 radians because

Converting from Radians to Degrees:

Since radians and 180° both measure the same angle, we can see that rad = 180°. This means that

1° = rad and 1 rad = {}°

This means that:

To convert from radians to degrees, we multiply by

For example, /18 radians is converted to 10°, because