Angle Conversions
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How to convert degrees to radians or radians to degrees.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to convert degrees to radians or radians to degrees.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

What are 'radians' ?

One radian is the angle of an arc created by wrapping the radius of a circle around its circumference.

In this diagram, the radius has been wrapped around the circumference to create an angle of 1 radian. The pink lines show the radius being moved from the inside of the circle to the outside:

           

The radius 'r' fits around the circumference of a circle exactly 2p times. That is why the circumference of a circle is given by:

            circumference = 2pr

So there are 2p radians in a complete circle, and p radians in a half circle.

Converting radians to degrees:

To convert radians to degrees, we make use of the fact that p radians equals one half circle, or 180.

This means that if we divide radians by p, the answer is the number of half circles. Multiplying this by 180 will tell us the answer in degrees.

So, to convert radians to degrees, multiply by 180/p, like this:

                                   

Converting degrees to radians:

To convert degrees to radians, first find the number of half circles in the answer by dividing by 180. But each half circle equals p radians, so multiply the number of half circles by p.

So, to convert degrees to radians, multiply by p/180, like this:

                           

Method:

Maths Helper Plus can show you the working steps for converting your angles between radians and degrees. It also draws the angle as an arc of a circle.  

Step 1 Download the free support file... We have created a Maths Helper Plus document to complement this topic. You can use it to practice the steps described below, and as a starting point for solving your own problems.

File name:  'Angle Converter.mhp'   File size: 7kb
Click here to download the file.

If you choose 'Open this file from its current location', then Maths Helper Plus should open the document immediately. If not, try the other option: 'Save this file to disk', then run Maths Helper Plus and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.

 

Step 2  Enter the angle to convert

Press the F5 key to display the parameters box:

 

The number in the 'X' edit box changes the type of conversion performed.

If 'X' is 1, then the 'A' value is converted from degrees to radians.

If 'X' is 2, then the 'A' value is converted from radians to degrees.

If necessary, click on the 'X' edit box and set the value to 1 or 2, depending on the type of conversion you want to perform.

Click on the 'A' edit box and enter the angle to convert. For degrees, just type a number. For radians, you can type a number, or you can enter a value in terms of p by using 'pi' for p.

 

Click the 'Update' button to update the calculations.

 

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