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A Bit About Radians

Updated on October 5, 2016

Do you remember looking at the different angular laws in geometry? Things like:

  • The angles in a triangle add up to 180°.
  • Parallel angles are equal.
  • Opposite angles in a parallelogram sum to 180°.

And if you were an enthusiastic student:

  • The angles in a polygon sum to 180(n-2)°.

Chances are you only worked with these rules in problems using the system of angular measurement that uses degrees as the unit. This system of measurement is so old we don't even know why it was invented, though some suspect it may be due to the Babylonians (Who loved a cheeky bit of mathematics.) using a sexagesimal number system; where instead of counting up to 9 and then adding a second column, they would have had to count up to 59 before adding a new column! That is, they would have to add a new column if they used the Hindu-Arab numeral system like us, which they didn't.


Anyway, back to degrees! As I was saying before, you likely used degrees throughout your school life. In the cases where you have had to record and work with angles outside of school and/or at work, you probably also use degrees. Degrees are not the only option however, there are actually two more systems of angular measurement used around the globe.

The Gradian

Gradians (Or the grad/grade.) is a system of angular measurement that came out of France during the introduction of the metric system. The gradian system defined a right-angle (90° turn.) as 100 grades, or 100g. The gradian never really caught on as well as the other units of the metric system did, and so is rarely ever used anymore, though they still see use in certain fields such as surveying and in the French artillery, according to some answers on forums.


The Radian

My personal (And many mathematicians) favourite system of angular measure. The radian is the most natural of all the measurement systems, as it does not rely on arbitrary division division. Instead, 1 radian is defined as the angle created by an arc of length equal to the radius of the circle itself. It sounds more complicated than degrees or gradians when phrased that way, but it is much easier to understand with a diagram:


Imagine you had a circle with a line representing the radius connecting the centre of the circle to its circumference, a radian would be the angle created by duplicated that line, wrapping it around the perimeter beginning where the other radius line met the perimeter, and then popping another radius line at the other end of the radian.

So, how many radians can you fit in a whole circle? You can fit exactly 2π radians inside a whole circle, meaning a complete (360°) turn is 2π radians, or 2πc, or 2πrads, whichever you prefer.

Common sense then tells us that in half a circle (Or 180°, half of a complete turn.) you can fit π radians. A right-angle (90°) turn is then π½ radians!


So how do we know that despite the size of the circle, there will always be exactly 2π radians? Well, radians as a form of angular measurement are in part derived from the formula for calculating the circumference of a circle:

C = 2πr (where C is circumference and r is radius)

Suppose we wanted to invent a system wherein angles are measured based on the radius of a circle, we might think to ourselves that it'd be a good idea to try dividing circumference by radius~

C/r = 2π

So using the formula for circumference, you can prove that for all circles, you can fit the radius r (Whatever that may be.) of that circle exactly 2π times into the circumference C!

By using the properties of a circle and completely avoiding any arbitrarily chosen divisions, we have created a pure, natural way of measuring angles. In fact, calculus wouldn't be possible if it weren't for radians.

Radian Measure
Degree Measure

Converting between degrees and radians is pretty easy. Recall how 360° is 2π radians, and that 180º is π radians? Well, using the fact that π radians = 180º, we can say that to convert θ degrees to radians, we can use the formula:

(Angle in radians) = θ(π/180) (Where θ is the angle is degrees)

And from radians to degrees:

(Angle in degrees) = θ(180/π) (Where θ is the angle in radians)

The Steradian

The steradian is a measure of angles in three-dimensions (Or a solid angle.). It is defined as being the solid angle subtended by a face of r2 on a sphere of radius r. The steradian is a lot more complicated and painful than the radian, but retains all the things that make radians useful and pretty amazing. A sphere is 4π steradians, and this remains constant no matter the size of the sphere. Again, like with radians, we can figure out why by looking at the formula for the surface area of a sphere:

A = 4πr2 (Where A is surface area and r is radius.)

With radians we wanted to divide the circumference up by the radius, so with steradians we'll cut the area up by the radius squared:

A/r2 = 4π

We see once again that thanks to the properties of a sphere, it can always be divided up into 4π steradians. If things like radians interests you, I recommend looking more into steradians (And quarternions if you're feeling brave.), or even buying a radian protractor if you are committed to shunning arbitrary divisions in measurement ;).


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