# Degrees in a Circle – Why 360?

The question came up on twitter about how we ended up with 360 degrees in a circle. My friend, Barbara commented that it probably had something to do with the Babylonian’s base 60 number system.

### Where did they get 360 degrees?

There are many speculations on how they came up with 360° in a circle. One of them, which I choose not to believe, is that there are roughly 360 days in a year.

The one that I like to believe has to do with hexagons and circles.

If you inscribe a hexagon inside a circle some neat things happen. For one, the perimeter of the hexagon is six times the radius of the circle. Check out the pinnable image to the right to see the details.

Since hexagons and circles go so nicely together, the Babylonians probably felt this was a great place to start in measuring angles around the circle.

The Babylonians use a base 60 number system. Which means instead of their place values being 10, 100, 1000, etc., their place values were 60, 602, 603, etc.

Just as we would naturally break things into 10 pieces, they naturally broke things into 60 pieces.

So from the center of the circle going around they broke each of the six equilateral triangles into 60 wedges. And since a hexagon has six equilateral triangles, that’s 6 x 60 = 360 wedges.

### Use it to teach math.

Give your students two regular hexagons of the same size. Have them cut one of them into six equal triangles.

Download instructions and hexagon template here.

If they don’t already know to cut from corner to corner through the center, this might be a nice discovery activity.

Discuss the the triangles, encouraging them to notice:

1. All the triangles are the same.
2. There are six triangles – one for each side of the hexagon.
3. The triangles are equilateral.

Now have your students draw a circle whose radius is the length of the sides of the triangles. Ask them to play with the triangles, the hexagon and the circle and see what they notice – they can stack them, see which goes inside which, etc.

Have a discussion about this. Encourage them to notice:

1. The hexagon fits exactly inside the circle (so the circle is circumscribed outside the hexagon, and the hexagon is inscribed inside the circle).
2. The six triangles, reassembled as a hexagon, also fit inside the circle.
3. The perimeter of the hexagon is six times the length of the radius.

### Let them create degrees!

Ask your students to pick their favorite number. They’re going to “wedge” one of the corners of their triangles into that many pieces. (I used 4 as my favorite number for the example in the picture below.)

Have them do that to one corner of each of the other triangles as well. Then place all the triangles inside the circle.

Now – how many “wedges” are there in the circle with all the triangles? For mine, it’s 6 x 4 = 24.

Let them know that the people who created degrees – the Babylonians – were in love with the number 60. So they did 60 wedges in each triangle! And that’s how we get 6 x 60=360 degrees!

Thoughts? Share them in the comments!

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### 4 Responses to Degrees in a Circle – Why 360?

1. You can share that idea with your students because it is geometrically pretty, and you can choose to like any historical explanation, but keep in mind that the Summarians used sexgesimal notation well before the Babylonians (from whom they obtained it).
More importantly, the Chaldeans routinely made and preserved examples of solar and lunar eclipses dating as early as 652 BC. They also calculated the recurrence of these events in periods, synodic months, with a span of about 29 1/4 days. By the first century BC they had conceived the zodiac recognizing major star patterns that moved in approximately monthly cycles across the heavens. Hence a period of 12 synoptic months.
The base sixty system was coincident with the emergence of the 360 day year, but your suggestion would have more likely led to a zodiac like method of 6 periods, not twelve.

• Bon says:

Wow, Pat. Thanks for the information!

2. The Babylonians knew the length of the year to be 365.25 days so anyone who claims that days-in-a-year had anything to do with it is a moron. They were quite competent at mathematics.

Far more likely is the idea that 360 is a really nice number, a highly composite number.

For people who worked with fractions instead of decimals, and who needed to subdivide a circle into many different sizes of piece (24) with integer sizes, 360 is the best choice.

• Bon says:

I’m not sure if they should be called morons…

But since you are the Curmudgeon, we’ll run with it. 😉

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