Why arc divided by radius is the measurement the mathematics prefers

0:00Last time we settled on three hundred and sixty, and we settled on it for a reason. It breaks into more whole pieces than anything near it. But a reason is not the same thing as a requirement. Nothing about a circle produces that number. Draw one circle cut into three hundred and sixty parts, and another cut into four hundred. The circle does not prefer either one. Both are decisions somebody made and everybody else agreed to.

0:33So here is the question worth asking. What could a circle supply by itself? Not a number we bring to it. A number it already has. The answer is sitting in the picture, and it has been there the whole time. Every way of measuring an angle so far has asked the same question. What fraction of the way round did you go? A right angle is a quarter of the way. A straight angle is half of the way.

1:03But a fraction of a whole turn needs somebody to have decided what the whole turn is worth. Three hundred and sixty. Four hundred. Anything at all. So stop asking for the fraction. Ask for the distance. When the ray turned, a point on the rim travelled. It swept out a piece of the rim, and that piece has a length. You could lay a string along it and read the length straight off.

1:31Nobody decided that length. The turning produced it. A length on its own is still not an angle. A bigger circle gives a longer sweep for exactly the same turning. So the arc has to be compared against something the same circle already owns. And a circle has exactly one length to offer. Its radius. Start with the simplest case. A circle whose radius is one. Turn the ray until the arc it has swept is also one.

2:04The same length as the radius. The angle you have just made is one radian. That is the entire definition. An arc as long as the radius. No unit was chosen and no number was invented. The circle handed over its own radius, and we used it as the ruler. Run the same recipe the other way and the sign comes back. Turn the ray clockwise until the arc is one, and the measure is minus one.

2:37Turn further, until the arc is one and a half, and the measure is one and a half. Clockwise, minus one and a half. Four pictures. The same construction every time. Two of the four are negative. And the sign is doing exactly what it did before. It records which way, not how much. Strip the signs off and only two different sizes are left. One, and one and a half.

3:05The minus is a direction flag. The size is the arc. Now the objection. That was a circle of radius one. What about every other circle? Take a circle of radius two. Sweep an arc of length two. Take a circle of radius one hundred. Sweep an arc of length one hundred. Every single one of them is one radian. I checked nine different circles, with radii from three sevenths up to a hundred, and all nine agreed.

3:39And not by dividing. Any two were compared by proportion. This arc against that radius, that arc against this radius. Thirty-six pairs, and every pair matched. Here is the control that makes that mean something. Keep the arc fixed at one and change the circle instead. On radius one, two, three and four, that same arc gives four different angles. One, a half, a third, a quarter. Six pairs, and not one of them is the same angle.

4:14The arc alone decides nothing. The arc against its own radius decides everything. One radius-length of arc is one radian. So what about an arc that is longer than that? Lay radius-lengths along it, end to end, and count them. On a circle of radius one, an arc of seven takes seven of them, and nothing is left. On a circle of radius two, that same arc of seven takes three, with one unit of arc left over.

4:48On a circle of radius seven, an arc of a hundred takes fourteen, and two are left. Every whole one you lay down is one radian. The leftover is a fraction of one more. Count them and you have the angle. Which is the same as saying the angle is the arc divided by the radius. The counting is the argument. The formula is what falls out of it. Write that down and something quietly remarkable happens.

5:21The angle is a length divided by a length. Centimetres over centimetres. Inches over inches. Whatever you measured in appears on the top and on the bottom, and it cancels. What is left is a bare number. Not a number of anything. Just a number. That is why a radian is not a unit the way a centimetre is a unit. It is a name for a pure quantity, which is why it can be, and usually is, left off entirely.

5:53Turn the relation round and it gives you a length again. Radius times angle hands the arc back. A length times a bare number is a length. The bookkeeping works both ways. A degree measure cannot do that, and in a moment you will see the damage. Before the damage, the number everybody wants. What does a whole turn measure? Go all the way round the circle of radius one.

6:22The arc you have swept is the entire way round, and that length is two pi. Divide by the radius, which is one. A full turn is two pi radians. About six point two eight three two. Half of that is a straight angle. Pi, about three point one four one six. Half again is a right angle. Pi over two, about one point five seven zero eight. And this holds on every circle.

6:52On radius one hundred the way round is two pi hundreds, and dividing by a hundred leaves two pi again. I checked five circles, and the answer never moved. Nobody chose two pi. It is simply what the way round is. Now put one radian back into the old unit and look at what comes out. One radian is fifty-seven point two nine five seven degrees, and it goes on.

7:23That number is horrible. And a great many people take it as proof that the radian is just as arbitrary as the degree. It is the exact opposite. Fifty-seven point two nine five seven is what one radian looks like measured in an arbitrary unit. The ugliness belongs to the three hundred and sixty. Not to the radian. If the degree were the natural unit, that conversion would come out clean. It does not.

7:55One radian is a little under two thirds of a right angle. One and a half radians is eighty-five point nine four degrees, just short of a right angle. Those two are worth holding on to, because they are the two pictures your eye can check. Here is the damage. Radius times angle gives the arc. That relation works in radians and it works nowhere else. Put a sixty degree angle into it by writing sixty, and watch.

8:28Sixty degrees in radians is pi over three, which is about one point zero five. You wrote sixty instead. The arc you get back is more than fifty-seven times too long. Not quite sixty times. But very nearly. One number, in the wrong unit, and the answer is out by a factor of almost sixty. You can have the relation in degrees if you insist on it. It reads: arc equals two pi r, times the angle over three hundred and sixty.

9:02I checked four cases and both versions agree, so the degree form is not wrong. It just drags a two pi and a three hundred and sixty along with it, forever. Now the move that makes all of this pay. Take the circle of radius one and mark the point where the ray starts. Stand a straight number line against the circle, touching it there. Put zero at the touching point. One and two above it. Minus one and minus two below.

9:35Now roll the line onto the circle. The top half wraps anticlockwise. The bottom half wraps clockwise. Watch where each number lands. The point marked one lands where the arc from the start is one. That is the terminal side of one radian. The point marked minus one lands the other way round, at minus one radian. The point marked two pi lands back exactly where it started. And the point marked two pi plus one lands precisely where one landed. The same place, with one more loop behind it.

10:14That last one should feel familiar, because it is the picture that was missing the loops. Roll the first twenty-four whole numbers on, and watch them scatter. They spread six to a turn, across the first four turns. Six of them land in each quarter of the circle. Six, six, six and six. Not one of them lands on an axis. And no two of them land in the same place.

10:41Then something quietly beautiful. Ask which of the first sixty whole numbers comes closest to arriving back where it started. The answer is forty-four. Forty-four radians is seven complete turns, and it misses by under two hundredths of a radian. In the old unit, that miss is just over one degree. Seven turns is fourteen half-turns, and forty-four over fourteen is twenty-two over seven. Which is the fraction that sits just above pi and misses it by about a thousandth.

11:18The whole number that nearly closes the circle, and the fraction that nearly is pi, are one fact wearing two hats. Step back and see what the rolling actually established. Every real number on that line lands on an angle. Every angle is landed on by a real number. So the radian measures and the real numbers are not two things that happen to correspond. They are one set of objects under two names.

11:49And that is what all of it was for. Once an angle is a real number, a function can take an angle the way it takes any other input. You will not need a protractor, and you will not need a unit. You will need a number line, and you already have one of those. Three hundred and sixty was a decision. This is not a decision. This is what the circle was going to say if you let it finish.