What are “radians”? A simple explanation of radian vs. degree
A simple introduction to radians.
You may learn about radians for the first time in a trigonometry class. Simply, a radian is another way to measure an angle instead of using degrees. You are merely switching the unit of measurement. It is like measuring your height in centimeters instead of inches.
This simple introduction will explain what radians are and how to calculate them. We don’t use radians in our everyday lives but you don’t need to feel intimidated by this unit of measurement.
Let’s begin with the basic parts of a circle:
An arc can refer to a full circle, a portion of a circle, or even longer than a circle. In this example, our arc is drawn as a full circle. The circumference is the length of that circle.
The radius (abbreviated to just “r”) is a straight line from the center of the circle to its edge. Be careful not to confuse “radius” with our similar-sounding friend “radians.”
As a rule, the radius of a circle can be used to calculate the circumference of a circle. The formula is:
2πr = circumference
Keep this formula in mind because it will return when we begin to calculate radians.
Units of measurement: Degrees
Degrees (abbreviated to °) are another unit of measurement, each counted as 1 of 360 equal slices of a circle. A full circle would be a total of 360°.
Why are circles divided into 360 parts? 360 is used for historical, not mathematical, reasons. The ancient Egyptians historically measured circles as a sum of 360 units because their measurement system was based on units of 60.
Despite being mathematically arbitrary, degrees are familiar to us. We use the language of degrees everywhere in everyday life. Idioms such as “a 180° change” describe moving in the completely opposite direction. Getting a “360° view” of something refers to a thorough perspective and understanding. Or the news may report on NASA’s 360° panoramic cameras.
In contrast, radians are pretty specific to higher level math. Don’t let the novelty worry you though. A radian is just another way to measure the same angle. For example, you can measure a circle as being 360° or as being about 6.28 radians.
HOW DO YOU MEASURE THE RADIAN?
A radian unit of measurement is based on the radius (which is why their names are similar). Take the length of the radius and measure out an arc. Imagine another line connecting the center of the circle to the end of that arc. The interior angle created by those two lines is 1 radius.
If you recall, the circumference of a circle = 2πr. The full circle’s angle = 2πradians.
HOW DO YOU CALCULATE RADIANS FROM DEGREES?
The full circle’s angle can be measured as 360°. We just learned that it can also be measured as 2πradians. So, we can write this as:
360° = 2πRadians
Simplifying that equation gives us
180° = πradians
We can rearrange this formula to calculate the measurement we need.
180°/π = 1 radian
or put another way:
1° = π/180 radians
Sometimes you may see unit circles annotated like this:
There are far too many formulas there to memorize. As long as you remember the formula of
circumference = 2πr
you will be able to calculate between degrees and radians for the entire unit circle.
Check out our math series on algebra, expressions, geometry and our two part post on quadratic equations. Make sure to connect with an UPchieve tutor to go over any questions. Not sure if you qualify for UPchieve’s free, on-demand, live tutoring? Head to our student sign-up page and check if your zipcode qualifies.
Originally published at https://upchieve.org on April 19, 2021.
