Radians to Degrees Calculator (Angle)

Degrees as a unit of measurement have been around for thousands of years. About six thousand years in fact.

It was the Mesopotamians who used a system based on the number 60 and first divided the circle in 360 degrees.

The same system was then taken up by the Egyptians, so all of the measurements for the pyramids will have been calculated using degrees.

Why Use Degrees?

So why the did Mesopotamians use 360? The reason is that its a very easy number to divide up.

In fact, 360 is divisible by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360.

It has 24 factors in total; that is, 24 numbers that will divide into and result in a whole number. In comparison, the number 100 has only 9 factors.

If you're looking for a system for measuring angles, which will always be fractions of a circle, then making the total for a circle 360 degrees means it can be split up easily in many ways.

Radians only began to be used about 300 years ago, invented by a mathematician named Roger Cotes.

He described their derivation, although the name radians would not be applied for another hundred years.

Why Use Radians?

Whilst degrees were designed to help measure the size of angles based on the proportion of a circle that they represented, radians were designed to be ideal for use in representing the length of the arc that the angle makes.

The simplest way to describe this is to talk about a slice of pizza.

Degrees are made to be useful for measuring the size of the angle at the pointy end of your slice.

But radians are designed to be useful for working out the length of the curved side of your pizza. So if you want to know how long your stuffed crust is, radians is the one you need!

This is where things start to get tricky.

To work out how to convert between radians and degrees, you need to compare the same angle in degrees and radians.

The easiest choice of angle is a full circle. We have already seen that a full circle is 360 degrees. But what is it in radians?

The answer is actually impossible to write down, at least with numbers. The reason it is impossible is that the answer goes on forever.

So you could keep writing for your whole life and never get near finishing.

This is because in a full circle, the number of radians is 2 Pi.

How To Convert

Technically, if we want to convert radians to degrees, we need to multiply by 360 and divide by 2 Pi.

But if we simply that, we can multiply by 180 and divide by Pi.

Degrees = Radians x 180 / Pi

Working the other direction, we can convert from degrees to radians in a similar manner.

Since 360 degrees equals 2 Pi radians, it must mean that 180 degrees are equal to Pi radians (by halving both angles).

Radians = Degrees / 180 x Pi

So to convert from degrees to radians, you would divide by 180 and then multiply by Pi.

I know what you're thinking: That's not even a number! And technically you're right. Pi (pronounced pie) is a Greek letter, that is used to represent a very special value.

Why Use Pi?

When you divide the length of the outside of a circle (the crust on our pizza) and divide it by the diameter (how wide the pizza is from one side to the other) you get the same answer every time.

No matter how big or small your pizza is. The answer begins 3.14159... but the digits actually continue forever.

This strange number that never ends is found in lots of places in nature and is one of the most important values in our universe. Unfortunately, it's a really horrible number to do sums with.

This is where radians come in.

By making the number of radians in a circle 2 times the value of Pi, it makes your calculations much simpler when you're trying to find lengths rather than angles.