# Circles and PiDegrees and Radians

So far in geometry, we've always measured angles in **full circle** rotation is **half circle** is **quarter circle** is

The number 360 is very convenient because it is divisible by so many other numbers: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, and so on. This means that many fractions of one circle are also whole numbers. But have you ever wondered where the number 360 comes from?

As it happens, 360 degrees are one of the oldest concepts in mathematics we still use today. They were developed in ancient Babylon, more than 5000 years ago!

At that time, one of the most important applications of mathematics was in astronomy. The *sun* determines the four seasons, which farmers have to know about when growing crops. Similarly, the *moon* determines the tides, which was important for fishers. People also studied the stars to predict the future, or to communicate with gods.

Astronomers noticed that the constellations visible at a specific time during the night shifted a tiny bit every day – until, after approximately 360 days, they had rotated back to their starting point. And this might have been the reason why they divided the circle into 360 degrees.

Of course, there are actually 365 days in one year (well, 365.242199 to be exact), but Babylonian mathematicians worked with simple sundials, and this approximation was perfectly adequate.

It also worked well with their existing base-60 number system (since

For many of us, measuring angles in degrees is second nature: there is 360° video, skateboarders can pull 540s, and someone changing their decision might make a 180° turn.

But from a mathematical point of view, the choice of 360 is completely arbitrary. If we were living on Mars, a circle might have 670°, and a year on Jupiter even has 10,475 days.

## Radians

Rather than dividing a circle into some number of segments (like 360 degrees), mathematicians often prefer to measure angles using the **unit circle**

A full circle has circumference

For a half circle rotation, the corresponding distance along the circumference is

For a quarter circle rotation, the distance along the circumference is

And so on: this way of measuring angles is called **radians**

Every angle in degrees has an equivalent size in radians. Converting between the two is very easy – just like you can convert between other units like meters and kilometers, or Celsius and Fahrenheit:

** 360°** = ** 2 π rad**

** 1°** = ** rad**

** 1 rad** = ** °**

You can write the radians value either as a multiple of *π*, or as just a single decimal number. Can you fill in this table of equivalent angle sizes in degrees and radians?

degrees | 0 | 60 | 180 | ||

radians | 0 | 2 |

## Distance Travelled

You can think of radians as the “distance traveled” along the circumference of a unit circle. This is particularly useful when working with objects that are moving on a circular path.

For example, the **speed of rotation** is

In a *actual* speed, because the length of the circumference is the same as one full rotation in radians (both are

The radius of the ISS orbit is 6800 km, which means that the *actual* speed of the ISS has to be

Can you see that, in this example, radians are a much more convenient unit than degrees? Once we now the speed of rotation, we simply have to multiply by the radius to get the actual speed.

Here is another example: your car has wheels with radius 0.25 m. If you’re driving at a speed of 20 m/s, the wheels of your car rotate at

## Trigonometry

For most simple geometry problems, degrees and radians are completely interchangeable – you can either pick which one you prefer, or a question might tell you which unit to give your answer in. However, once you study more advanced

Most calculators have a **sin****cos****tan** take angles as input, and their inverse functions **arcsin**, **arccos** and **arctan** return angles as output. The current calculator setting determines which units are used for these angles.

Try using this calculator to calculate that

sin(30°) =

sin(30 rad) =

Using radians has one particularly interesting advantage when using the **Sine function**

sin(

This is called the **small angle approximation**, and it can greatly simplify certain equations containing trigonometric functions. You’ll learn much more about this in the future.