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How to Change Degrees to Radians

Many math problems require you to change measurements from degrees to radians. Degree measures are more familiar than radians to most people. A circle is divided into 360 degrees, and you have a pretty good idea what angles of varying degree measures look like. Radians are another story.

A circle is also divided into slightly more than six radians — a radian is much larger than a degree. Radians are very useful when doing computations involving other numbers, because radians are the same type of number. Degrees are of a different type or category.

You often perform mathematical computations in radians, but then convert the final answers to degrees so the answers are easier to visualize and comprehend. You can use a nifty little proportion to change from degrees to radians or vice versa.

In this proportion, the Greek letter theta, θ, represents the name of the angle. Putting the superscripts ° and R on θ makes the angle stand for the measure in degrees and radians, respectively.

image0.png

This proportion reads: "The measure of angle θ in degrees divided by 180 is equal to the measure of angle θ in radians divided by π." Remember that π is about 3.141592654.

The computation required for changing degrees to radians isn’t difficult. It has a few tricks, though, and the format is important. You don't usually write the radian measures with decimal values unless you've multiplied through by the decimal equivalent for π.

To change a measure in degrees to radians, start with the basic proportion for the equivalent angle measures:

image1.png

For example, here’s how you change a measure of 40 degrees to radians:

  1. Put the 40 in place of the θ0 in the proportion.

    image2.png
  2. Reduce the fraction on the left.

    image3.png
  3. Multiply each side of the proportion by π.

    image4.png
  4. Simplify the work.

    image5.png

Check out another example: Change a measure of –36 degrees to radians.

  1. Put the –36 in place of the θ0 in the proportion.

    image6.png
  2. Reduce the fraction on the left.

    image7.png
  3. Multiply each side of the proportion by π.

    image8.png
  4. Simplify the work.

    image9.png

Having a negative angle is fine. You leave the expression as a fraction; don't change it to a decimal form.

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