The degree: why a full turn was cut into 360 equal parts

0:00Last two videos, we got quite good at comparing angles. We can say this one is bigger than that one. We can say these two are equal, and prove it by folding. And we did all of that without a single number, which I think is worth being slightly proud of. But there's a question we still can't answer, and you can feel the gap. Which is bigger? We can do that. By how much? Nothing. Silence.

0:28Comparing puts angles in order. It never tells you the size of the gap between them. And you can't write down an angle, or send it to somebody, or check it against a specification, with only an order. For that you need a number. So this video is about where the number comes from. And here's the thing I want you watching for: nobody found it. Somebody made it. Go back to the circle we used for comparing. The blank one, with nothing written on it.

1:02It's still a good idea. We put its centre on the vertex, and the angle became two marks on the rim. What we're going to do now is put marks on that rim, so that we can count instead of compare. And notice what kind of ruler this is. An ordinary ruler measures length, and it's a straight thing you lay alongside another straight thing. This one measures turn. So it's round, and you put its centre on the corner.

1:31Same idea, different quantity. Length gets a straight ruler. Turn gets a round one. So here's the move, and it's the whole of the video in one sentence. Take one full turn — all the way round, back to where you started. And cut it, at the centre, into three hundred and sixty equal pieces. Not roughly equal. Exactly equal, all three hundred and sixty of them. Now pull one of those pieces out and look at it. It's a very thin wedge.

2:06That piece is called one degree, and we write it with a little circle raised up after the number. And that's it. That's the definition. A degree is one three-hundred-and-sixtieth of a full turn. It didn't come from anywhere else. It isn't borrowed from lengths. It's a unit of turning, and we manufactured it by slicing. And now measuring an angle becomes something almost dull, which is exactly what you want from a unit.

2:38Take an angle. Put the centre of the marked circle on its vertex. Now count how many of those thin wedges fit inside it. Your book draws one where the answer is thirty. Thirty unit parts, packed in between the two arms. So we say the angle measures thirty degrees. That's all a measurement is. It's a count of unit parts. Nothing cleverer than that. And notice this kills a misconception before it starts.

3:08The degree isn't a distance along the rim. Draw a bigger circle on the same angle — the rim is longer now, but the count is exactly the same. Same thirty wedges. Wider ones. The turn didn't change, so the measure didn't change. Which is the arm-length lesson again, wearing a different coat. Now let's give our two landmarks their numbers, and notice that we don't have to be told them.

3:38A full turn is the whole thing, all three hundred and sixty pieces. So a full turn is three hundred and sixty degrees. A straight angle is half a turn. So take half of three hundred and sixty. One hundred and eighty. A straight angle is a hundred and eighty degrees. And a right angle is half a straight angle — that was last video's definition, proved by folding. So take half of a hundred and eighty. Ninety.

4:10A right angle is ninety degrees. And I want you to see that you derived those. Nobody handed you ninety. It fell out of two halvings. Which means if you ever forget one, you can rebuild it from the full turn in about four seconds. Right. The obvious question. Why three hundred and sixty? Why not a hundred? Why not a thousand? Both would be easier to say. Your book has a section on this, and it gives you the historical material honestly.

4:44There are ancient calendars that counted a year as three hundred and sixty days. There's a line your book quotes from the Rigveda, describing a wheel with three hundred and sixty spokes. And there were Babylonian mathematicians who counted in sixties rather than in tens — sexagesimal, it's called. Sixty times six is three hundred and sixty. So the number is old, it shows up in several places, and it's clearly tangled up with the year and with counting by sixties.

5:21But here's the part I want you to hear clearly, because your book is unusually straight about it. Nobody fully knows. There isn't a proof. There's no property of circles that forces the number three hundred and sixty. If someone tells you there are three hundred and sixty degrees in a circle because of something about circles — they're making it up. It's a human choice. A very old one, made by people we can't fully identify, for reasons we can only partly reconstruct.

5:56And I think that's a better thing to know than a tidy invented story would be. But there is a second answer to why three sixty, and this one we can check ourselves, right now, with division. Here's the practical reason the number survived. You very often want a simple fraction of a circle. A half. A third. A quarter. A fifth. A sixth. And you'd rather those came out as whole numbers of units, instead of something with a decimal trailing off.

6:31So let's test it. Divide three hundred and sixty by everything from one to ten. By one: three hundred and sixty. By two: one hundred and eighty. By three: one hundred and twenty. By four: ninety. By five: seventy-two. By six: sixty. By seven: and this one fails. Three hundred and sixty divided by seven isn't a whole number. Keep going. By eight: forty-five. By nine: forty. By ten: thirty-six.

7:04So every number from one to ten divides it cleanly except seven. And don't quietly drop the seven, because that would make the claim false. It's nine out of ten, not ten out of ten. It also divides by twelve, giving thirty. And by twenty-four, giving fifteen — which is handy if you're carving up a day. Your book's claim is that no smaller count does all of that, and I checked it: the smallest number divisible by one through ten, leaving out the seven, is exactly three hundred and sixty.

7:42Not approximately. Exactly. Three hundred and sixty is the smallest number that works. Your book then gives you ten circles, each already cut into equal slices, and asks you for the size of one slice. And now you can do all ten in your head, because it's just division. One piece: the whole turn, three hundred and sixty degrees. Two pieces: a hundred and eighty each. That's the straight angle, and it should be.

8:14Three: a hundred and twenty. Four: ninety — there's the right angle. Five: seventy-two. Six: sixty. Eight: forty-five. Nine: forty. Ten: thirty-six. And twelve: thirty degrees. Which is the one on a clock face, between one hour mark and the next. Notice you never measured anything there. You divided. That's what having a unit buys you — questions about turning become questions about arithmetic. Now, the best thing in this section, and you should genuinely do it rather than watch it.

8:54Your book has you build the protractor yourself, out of paper, using nothing but folding. Cut a circle out. Any size, it doesn't matter — and by now you know why it doesn't matter. Fold it exactly in half and cut along the fold. You've got a semicircle. Its straight edge is a straight angle, so write zero at one end and a hundred and eighty at the other. Now fold the semicircle in half, bringing one end of the straight edge onto the other.

9:26The crease points straight up. And that crease cuts a hundred and eighty into two equal parts. Half of a hundred and eighty is ninety. So write ninety on that crease. You just made a right angle out of a piece of paper, with no instrument at all. And now don't stop, because the pattern is the point. Fold again. Bring the zero edge onto the ninety crease. That new crease sits halfway between zero and ninety. Half of ninety is forty-five.

9:59Do the same on the other side and you get the crease halfway between ninety and a hundred and eighty, which is a hundred and thirty-five. Fold once more, and each of those halves again. Half of forty-five is twenty-two point five. And notice — that isn't a whole number, and your book doesn't round it. It writes twenty-two point five. Because halving doesn't care whether you find the answer tidy.

10:28Open the paper out. You now have creases at zero, twenty-two point five, forty-five, sixty-seven point five, ninety, a hundred and twelve point five, a hundred and thirty-five, a hundred and fifty-seven point five, and a hundred and eighty. Nine creases, eight equal gaps, filling one straight angle. And your book asks why each gap is twenty-two point five. There's your answer: a hundred and eighty divided by eight. One more fold would give eleven point two five degrees, and at that point the paper gets thick and the creases get vague. That's roughly where making it yourself stops and buying one starts.

11:16One last word, and you've been doing it all video without naming it. Every time you folded, you took one angle and made two equal angles out of it. That has a name. It's called bisecting the angle. Cutting it into two equal parts. And the crease that does the cutting is called the angle bisector. Notice the word equal is doing the work again. Bisecting isn't cutting an angle into two pieces — any ray does that, we saw it last video.

11:49It's cutting it into two equal pieces, and the fold is what guarantees they're equal. So look at what you've built. Every mark on your paper protractor is a bisector of the one before it. The printed one in your geometry box is the same object, with more marks and better manufacturing. The tool didn't create the unit. The unit came first, from one decision — cut the turn into three hundred and sixty — and everything after that is counting and halving.

12:24Here's yours. You've got creases every twenty-two point five degrees. Using only those creases and the two edges, which angles can you make exactly? And can you get sixty degrees out of a paper protractor made only by folding? Have a think about what folding can and can't reach. Comments below. Next time: we take a real protractor, and read an angle off it — including the trap of the two scales running in opposite directions.