Converting Radians to Degrees

Written by tutor Yvonne H.

If you are wondering how to convert from radians to degrees, then you probably have some experience converting from one
unit to another. For example, you probably know how to measure the length of your bedroom in feet and convert those feet
into inches, i.e. 18 feet = 18 X 12 = 216 inches long. The same concept, unit conversion, applies to converting from
radians to degrees and from degrees to radians. Keep in mind that unit conversion means that there is a ratio that defines
the relationship between one unit to another, such as the ratio of 12 inches to 1 foot (12 in: 1 ft, or written as 12 in/1 ft).
This ratio is called the conversion ratio.

Before we convert from radians to degrees, or vice versa, let’s look at the measurement of a circle in each unit. In other
words, let’s measure a circle in degrees and again in radians. As we measure the circles, notice the difference in notation
and try to determine the simplest ratio that relates the units, radians and degrees.

Notice that when we measured in degrees we use the degree symbol ,°, but when we measure in radians, we use the symbol for
the number pi, π. Now we can figure out what the conversion ratio is between degrees and radians. Let’s do that together.

Question 1: Which degree measure coincides with a whole π?

In other words, look at the position of a whole π in Figure 2. Now find the same position but in the circle in Figure 1. There
is the answer for Question 1, 180°. Great! Now we have our conversion ratio, π = 180°. You can use this ratio to convert from
radians to degrees and from degrees to radians. Are you ready to try some practice exercises? Let’s try the first one together.

1. Here is an attempt to convert 2° to radians. Why is this calculation wrong?

2° × ^180°/_π = ^360°/_π

Here is a clue:

5 ft × ^{12 in}/_1ft = 60 in

In the conversion from 5 feet to inches, the unit feet canceled out because multiplication allows us to do that for any values that are
the same in the numerator and denominator. Remember, this does not work for addition and subtraction, only multiplication. After the
unit, “feet,” was canceled out, we were left with only the unit we wanted, namely inches.

This means that we need to flip the conversion ratio to correct the calculation from 2° to radians.

Correction:

2° × ^π/_180° = ^π/_90°

Now the unit degrees, cancels out and we are left with the unit radians.

Radians and Degrees Practice Quiz