Have you ever thought about how it’s kind of weird that a circle has 360 degrees? At first thought, it seems like a rather random number to have chosen—why not 100, or 500, or 720 degrees? Was it really a random choice? Or was there actually some good reason that 360 was chosen to be the number of divisions in a circle?
As we’ll find out today, there was indeed a good reason. What was it? We’re not entirely sure. But we do have some pretty good ideas. And those ideas are exactly what we’re going to be talking about.
The When and Where of 360 Degrees
As you probably know, these days we humans like to divide a circle up into 360 pie-shaped wedges. Each of these wedges contains an angle at its vertex, and we say that the size of this angle is 1 degree. As you also probably know, degrees aren’t the only way we can measure angles. Angles are also measured in radians and sometimes (very rarely), they are even measured in obscure military units called gradians (which is why a lot of calculators have “deg rad grad” buttons on them).
While we don’t know exactly why the 360 degree convention was chosen (more on that in a minute), we do know approximately when and where it all started. At least we know that it came to be a long, long time ago—as in 4 or 5 thousand years ago with the Babylonians, the Greeks, and perhaps other even more ancient groups.
As to the question of why 360 degrees was chosen, here’s what we think happened …
Reason #1: The Length of the Year
Even if you have absolutely no idea right at this instant why there are 360 degrees in a circle, I bet that if you stop and think for a few minutes you can figure out one possibility. If after those few minutes you’re still not sure, think about where else you’ve seen a number that’s close to 360 in your life. And if you’re still stuck after that, think about the Sun … the Earth … orbits … and calendars.
You might conclude that the Sun moves about 1/360 of the way along this circle every day.
The Earth takes one year to orbit the Sun. And a year is just a little more than 365 days. That means that the Earth rotates on its axis a little more than 365 times every year. And it means that every day the Sun appears to move about 1/365 of the way along a huge circle projected onto the sky that extends all the way around the Earth (called the ecliptic). If you lived a few millennia ago and didn’t have modern instruments to accurately record the positions of objects in the sky, you might conclude that the Sun moves about 1/360 of the way along this circle every day, which is exactly what ancient astronomers did. And they then made a leap and decided to divide this circle on the sky—and all circles—into 360 even parts so that the Sun would move through 1 part per day. Each of these parts was dubbed 1 degree, thus giving us the idea that a circle contains 360 degrees.
Makes sense, right? And given that the ancient Babylonian and Persian calendars were both based upon 360-day years, it seems likely that this simple astronomical observation is the reason a circle contains 360 degrees.
Reason #2: Babylonians and Base-60 Numbers
But that’s not the end of the story. Because there are other reasonable ideas out there as to the origin of the 360 degree convention. As we saw earlier, the Babylonians used a 360 day calendar. And, as it turns out, the Babylonians also used a base-60 number system (called the sexagesimal system). Just as we use 10 different symbols to represent numbers in our base 10 decimal system, the ancient Babylonians used 60 symbols to represent numbers.
Why does this matter? Well, 60 x 6 = 360. This means that 360 is a nice even multiple of the number base in the Babylonian system (which would have had the same aesthetic value to their brains that a nice even multiple of 10 has to ours). But there’s more to it than that. The Babylonians knew about equilateral triangles. And they knew that if you arranged 6 of these equilateral triangles in a certain way with the edge of one aligned on top of the edge of the next, the last one would end up meeting back up with the first. In other words, the total angle formed by 6 of these equilateral triangles would be the same as the angle around a circle. Given the Babylonian usage of 60 as their number base, they decided that each of the angles of an equilateral triangle would be 60 degrees. And thus, when you multiply these 60 degrees by the 6 equilateral triangles that combine to create a sort of circle, you get 6 x 60 = 360 degrees. And thus, 360 degrees in a circle.
So, there’s that.
Reason #3: The Many Factors of 360
But that’s still not the end of the story … because there’s another reason to love the number 360. Namely, it’s evenly divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180. That’s a lot of factors!
And that makes 360 a really convenient number because it means we can divide a circle into 2, 3, 4, 5, 6, 8, 9, 10, 12, and so on even parts. It makes solving problems by hand—which, mind you, was the only way to solve problems thousands of years ago—much easier.
While this alone doesn’t seem like enough reason to have swayed people to define a circle as having 360 degrees, it certainly wouldn’t have hurt. And it’s entirely possible that it was a combination of all three reasons (and possibly others as well) that ultimately lead us to the definition of a degree that we still use today.
OK, that’s all the ancient math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Earth and Sun image from Shutterstock.