## Why are there in a circle?

I would imagine all of you reading this know that when we learn about angles, we learn that we measure angles in degrees, and that there are in a full circle. Why this strange number?

It was the Babylionians who chose to have in a full circle. When you think about it, they could have chosen any figure, say 400 or 10 or 1,000. Why did they choose 360? Well, it turns out that 360 was a special number to the Babylonians, or to be more correct 12 and 30 were, and 360 is 12 multiplied by 30.

12 was special to the Babylonians because there are 12 and a bit cycles of the Moon in a year. It is from the time between e.g. each new Moon (or full Moon) that we get our idea of months. 30 was special to the Babylonians because this was, roughly speaking, the number of days in each lunar cycle.

So, if 12 and 30 were special, then the number produced by multiplying them together, 360, was even more special. That’s why we are stuck with the strange number for the degrees in a circle.

## Radians – more natural units

Mathematicians, physicists and a lot of other scientists most often use a different unit to measure angles, a *radian*. It is a more natural unit than degrees. A radian is very simply defined. If we have a circle with a radius , and we draw an angle as shown in the diagram below, then we will call the length of the arc produced .

The definition of the *radian* is quite simply that if is measured in radians then

## Converting between degrees and radians

Converting between these two ways of measuring an angle is quite simple. Remember that the circumference of a circle, the distance all the way around, is . But this is just the length of our arc when . So when this is equivalent to radians.

This table shows the conversion from degrees to radians for some commonly used angles.

Degrees |
Radians |

0 | 0 |

30 | |

45 | |

60 | |

90 | |

180 | |

270 | |

360 |

## The sine function

The sine function, , can be multiplied by a factor to produce a regularly varying function of any amplitude. For example, would look like this (remember we are expressing in radians, not degrees).

The sine function is very useful in many areas of mathematics and physics.

## Time varying waves

Suppose we have a wave travelling out from a stone which has been dropped into the water in a smooth pond. The ripples (waves) will travel outwards from where the stone was dropped. If we look at the height of the water at some point on the surface of the pond, then we find that the height of the water will vary with time as . In this expression is the mean (average) height of the water (the level of the water before we dropped the stone), is the size of the wave (the peak height), is something called the *angular velocity*, is time (in seconds), and is something called the *phase angle*.

### Angular velocity

Linear velocity, the velocity of something moving in a straight line in, say, the x-direction is just defined as . If an object is moving in a circle, we can define a similar concept called the *angular velocity*, which is not the linear distance divided by time, but rather the *angular distance* divided by time. But, the angular distance is just (when expressed in radians), so the angular velocity and is measured in radians per second. Re-arranging this, we can see that , so in other words has the same units as radians, they multiply together to measure an angle. This means that is equivalent to , enabling us to consider the variation of a wave as a function of time.

When expressed in radians, the angle for one complete cycle of a sine curve is just given by . Using our relationship between and we can say that the time taken for one complete cycle, which we called the *period* is just given by . So, depending on the value of the angular velocity , this determines how many seconds it takes for the sine function to complete one cycle.

### The phase angle

The phase angle, above, is needed as our zero time may not be when the expression is zero. So, it just allows us to correct for when we start timing, just in case this is not at the moment when our sine function has zero value.

### Putting this together

Below are plots of two sine curves, the first (the blue curve) is given by . The second one (the green curve) is of (remember the phase angle is expressed in radians, and one radian is approximately .

Because , this means the period of the wave will be . You can see that the blue curve does indeed complete one cycle in 8 seconds. The green curve has the *same* value of , so it has the same period. However, it is shifted in time due to the phase angle . So, at a time of , the green curve is *not* zero, and we say that it is *ahead* or *leads* the blue curve as it reaches its first peak *before* the blue curve does. The number of seconds it leads the blue curve can be found by considering that so , which in this example is .

In tomorrow’s blog I will discuss an exam question I have been going over with my students, where the height of water in metres in a harbour is given by e.g. (where is expressed in hours). I will discuss how we can work out the maximum and minimum heights of the water, when e.g. the first maximum occurs, and how many hours a day we can use the harbour if we need a minimum of e.g. 2 metres of water in the harbour.

on 03/06/2013 at 05:10 |Safely into the harbour | thecuriousastronomer[…] « Degrees, radians, angular velocity and phase angle […]

on 24/10/2013 at 06:05 |A simple pendulum and circular motion | thecuriousastronomer[…] force, , can be written using Newton’s 2nd law as . The angle is measured in radians (see my blog here for a tutorial on radians). When is small, and so we can write […]

on 11/03/2014 at 08:02 |Derivation of the centripetal force | thecuriousastronomer[…] the arc-length, which we shall call is related to the angle via our definition of the radian (see this blog here), , so we can write The acceleration . But the speed and time are related via , so we can […]

on 02/10/2017 at 04:34 |The Beauty of PlayI’m teaching a group of homeschooling middle school children physics and this was useful for me. Thank you.

on 24/03/2018 at 22:48 |Stephen O'ShaughnessyI’m interested in creating my own website dedicated to math topics, and I’m interested in what illustration software you use to display content. I’m particularly interested in how you rendered your definition of the radian above.

on 25/03/2018 at 19:23 |RhEvansI used an app called “TouchDraw”

on 28/03/2018 at 03:06Stephen O'ShaughnessyCurios,

Thanks for the feedback. I’ve looked at some of the videos and will download TouchDraw on my Android tablet.

I’ve been experimenting with another software called Geometry Expressions, by Saltire. I like the fact that it has 3D capabilities (including transparency), can work in TeX formats, and can create HTML5/JavaScript apps ( https://www.youtube.com/watch?v=1PAXZTnSEuk ) .

Steve