I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.
360 Degrees in a Circle
As any schoolchild can tell you, there are 360 degrees in a full circle. By splitting a circle up into 360 equal slices we can then measure any angle of turn that we like, whether it be half a turn (180°), two complete turns (720°) or any other size of the angle. But why do we use 360°? Why did somebody pick this apparently random number and not something else like 10, 100 or even 540?
There are other systems such as radians (splitting a circle into 2π pieces) or even gradians (splitting a circle into 400 pieces), but it is still the 360 degrees that we tend to use in everyday life.
While nobody is sure of the exact origins of the use of 360, there are several hypotheses that we will have a look at here.
Reason 1 - Ancient Civilisations and the Sexagesimal System
One potential reason for the use of 360° is in the sexagesimal system used by the ancient Sumerians and Babylonians. Whereas we use the base-10 decimal system of numbering which features ten digits 0-9, the base-60 sexagesimal system uses sixty distinct symbols.
Just like our decimal system matching up with people having ten fingers, the sexagesimal system can also be linked to fingers and hands. Each finger has three knuckles (the joint where two-finger bones meet). Ignoring the thumb, this means that you have 3 × 4 = 12 knuckles on one hand. Whereas we may keep track of numbers by counting on our fingers, with each finger representing 1, we could count on our knuckles instead. By doing this we could get to 12 on one hand.
Once we reach 12, we can raise one finger on the other hand to represent the fact that we have already counted one set of knuckles and then start counting again on the first hand. As we raise one digit for each 12 knuckles that we count and there are five digits on one hand, we can get to 5 × 12 = 60 before running out of fingers and knuckles. In this way, the sexagesimal system is just as useful for human hands as the more familiar decimal system.
After the Sumerians and Babylonians, the sexagesimal system was then passed on to the ancient Egyptians who, with their love of geometry, realised that you could fit six equilateral triangles together into a full circle such as in the diagram below. The Egyptians had already decided that each angle of an equilateral triangle was their favourite number 60°. This then makes the full circle equal to 6 × 60° = 360°.
Reason 2 - The Calendar
Another potential source of our use of 360° is in the calendars of the ancient world. It has been known for many thousands of years that the Earth takes approximately 360 days to travel around the Sun. The ancient Persians, for example, had a 360-day calendar with an extra intercalation month added in to keep the calendar in line with the seasons (essentially matching our current 365-day calendar). If you have a 360-day calendar, then it is a logical step to assume that the Earth travels 1/360 of its orbit per day, hence splitting the circular orbit into 360 equal pieces.
Furthermore, if we take the average of the approximately 365-day solar year and the approximately 355-day lunar year, we again get 360 days.
Reason 3 - The Mathematics
The third reason for using 360 is that although at first glance it may seem like a random number and harder to use than a power of ten such as 10 or 100, it has one particularly useful property; its large number of factors (numbers that divide into another number perfectly without leaving a remainder).
Having a large number of factors means that 360 can be divided up extremely easily. If we want to split the circle into two, we can do 360º ÷ 2 = 180º, a whole number. If we wanted to split the circle into twelve parts we can do 360º ÷ 12 = 30º, another whole number.
In fact 360 has 24 factors; 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 itself. We could split a circle equally into any of these numbers without having to use decimals, fractions or remaining bits. If we compare this to other numbers such as 10 with its four factors and 100 with nine factors we can see just how easy it is to divide 360º into equal slices in different ways.
360 is known as a highly composite number. This means it has more factors than any number below it. If we wanted a number with even more factors, we would have to go all the way up to 720 which has 30 factors.
As we have already seen, there is more to the choice of 360 than just its factors, but this ease of division makes 360 an ideal number of degrees regardless of how this number was originally conceived.
Bibliography and Further Reading
- Science ABC - Why is a full circle 360 degrees instead of something more convenient like 100?
- Popular Science - Who determined that a circle should be divided into 360 degrees?
- Quick and Dirty Tips - Why does a circle have 360 degrees?
- Wikipedia - Babylonian Calendar and Lunar Calendar
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
© 2021 David