Degrees and Radians

There are two commonly used units of measurement for angles. The more familiar unit of measurement is that of degrees and the other one is radian.

Angles in degrees

A circle is divided into 360 equal degrees. Degrees may be further divided into minutes and seconds. Each degree is divided into 60 equal parts called minutes. Each minute is further divided into 60 equal parts called seconds.

For example, 34 degrees 26 minutes 51 seconds which can be written as 34° 26’ 51’’

Parts of a degree can also be referred to decimally. For example, 60 degrees 30 minutes is 60 and a half degrees which can be written as 60.5°

Angles in radians

Definition 1:

We can define radian in terms of the arc length and the radius:

The size of an angle in radian is given by the ratio of the arc length to the length of the radius.

An angle with 1 radian will have an arc length that is equal to the length of the radius.
An angle with 2 radians will have an arc length that is twice the length of the radius.

Definition 2:

We can also define radian in terms of the unit circle (a circle of radius 1).

Consider the unit circle whose center is the vertex of the angle to be measured. The angle cuts off an arc of the circle, and the length of that arc is the radian measure of the angle.

This second definition allows us to convert between degree measurement and radian measurement easily.

We know that the circumference of a circle is 2πr. For a unit circle, r = 1 and so the circumference = 2π

This means that 360° equals 2π radians.

Therefore, 1° = radians, and 1 radian = degrees.

The following are some common angles in both degree measurement and radian measurement.

Videos

The following video gives a basic introduction into degrees and radians and converting between them.

The following video gives more examples of converting angle measurements from degrees to radians.

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