When you are first introduced to angles, you typically refer to them in terms of “degrees.” Chances are, you know what a 45° angle looks like. Also, by the time you reach middle-school, you know that a triangle has 180°. It’s easy, cut and dry, black and white. However, most of the scientific geniuses of modern time opt to use radians to measure an angle.

Now the question that may come to mind is – why radians? Degrees are pretty easy, right?

The History of Radians

Back in the dark ages, the first mathematicians determined a degree’s size based on the divisions of a full circle. Degrees are the same thing as a slice (1/360) of a circle. While there is no real reason as to why a circle has 360°, it is something you must accept and move on from.

Regardless of the reason, 360 is a great number. This is because it is easily divided (evenly) by many other numbers – 180, 120, 90, 72, 60, 45, 40, 36, 30, 24, 18, 15, 12, 10, 9, 8, 6, 5, 4, 3, and 2. The earliest measures of distance and time relied on having easy and convenient numbers to work with.

Enter the Radian

Unfortunately, the radian isn’t as easy and nice since it isn’t a whole number. Some believe that radians were originally developed to provide mathematicians with a method to relate the measure of an angle to the size or radius of a circle.

Radians are much larger than degrees. Circles have 2π radians, which is just a little over six radians. A single radian is approximately 1/6 of a circle or just over 57°.

Why You Use Radians?

The biggest advantage offered by radians is that they are the natural measure for dividing a circle. If you take the radius of a given circle and bend it into an arc that lies on the circumference, you would need just over six of them to go completely around the circle. This is a fact that is true for ALL circles.

Converting between Degrees and Radians

If you are given degrees and want to convert it to radians, there is a simple formula to use:

radians which computes to (approximately) 0.017D radians

If you are given radians and want to change them to degrees, use the formula:

degrees which computers to (approximately) 57.3R degrees

The good news is, you don’t have to remember these formulas. Most calculators are equipped with functions that will help you convert from degrees to radians and back again. In the world of math, this will make it easier to come “full circle” with difficult math problems.

Favorite Radians

The most used or favorite angles are those that are a multiple of 15 degrees, such as 90, 60, 45, and 30 degrees. Thanks to the skill of early mathematicians (or pure luck – we can’t be sure) converting these angles into a radian measurement is quite simple. For example, 90° angles are equivalent to π/2; 60° angles are equivalent to π/3 radians; 45° angles are equivalent to π/4 radians; and 30° angles are equivalent to π/6 radians. The most frequently used radian measures are those that have denominators of six, four, three and two.

Now you are ready to work with radians. With the information here, you are well on your way to becoming a member of the mathematical elite.