Arc length as the central angle's share of the circumference

0:00There are two formulas for the distance round a circle, and it is not always clear which to reach for. One says it is pi times the width across. The other says it is two pi times the distance from the centre out. They are not two rules. The width across is two of those distances, so the second is the first with that substitution already made. The second is the one to keep, because everything that follows is about taking a share of it.

0:34Draw a line straight through the centre. It cuts the circle into two curved halves; colour them differently. Fold the page along that line and the top arc lands exactly on the bottom one. Folding does not stretch anything, so the two arcs are the same length. Two equal pieces make the whole way round, so each is half of it: pi times the radius. That is a proof, not a measurement.

1:02Now do it a second way. Instead of folding, spin the whole circle halfway round its centre. The top arc goes where the bottom one was, and turning does not stretch anything either. Same conclusion. It is worth having both: folding needs a line, so it handles halves and nothing else. Turning needs no line, and turning is what will keep going. Put in a second line through the centre, square to the first.

1:33Now there are four arcs. Spin the circle a quarter turn and every arc lands on the next one round. So all four are equal, and each is a quarter of the way round: pi times the radius, over two. Notice what did the work. It was not a fact about quarters. It was that turning a circle about its centre slides it onto itself and changes no length. Now rewrite both results in a way that looks like extra work.

2:06The half arc is pi times the radius. Write it instead as the whole way round, times a hundred and eighty over three hundred and sixty. The quarter becomes the whole way round, times ninety over three hundred and sixty. The numbers have not changed. What has changed is that the angle is now visible - a hundred and eighty degrees out of the three hundred and sixty in a full turn, and ninety out of three hundred and sixty.

2:37The arc is not computed from the angle by some rule. The arc is taking the angle's share. Once the fraction is on show, the general case writes itself. Take any angle at the centre and call it theta degrees. The two lines out to the edge cut off an arc, and that arc is the whole way round, times theta over three hundred and sixty. It is the same sentence with a letter where the number was.

3:09And this is where a lot of teaching stops: formula written down, two cases behind it. Two cases is not much. How much is two cases worth? Here is a different rule for arc length. At ninety degrees it is exactly right. At a hundred and eighty degrees it is exactly right. It agrees at both cases just proved, and it is nonsense: it says a sixty-degree arc is as long as a ninety-degree one.

3:43So two cases do not pin a formula down. Something else is doing the work, and it is this: arcs joined end to end add up. Lay a thirty-degree arc against another and you have the sixty-degree arc. The rogue rule breaks that at once - its two thirty-degree pieces come to twelve sevenths of the arc they are meant to make. Tested at every whole angle up to a hundred and seventy-nine, it fails a hundred and seventy-seven of them.

4:15It survives at ninety, where it was built to agree, and at a hundred and fifty by accident. The real rule passes all one hundred and seventy-nine. Turning and adding are enough on their own. Cut the full turn into some number of equal angles. Turning says the arcs they cut are all equal; adding says those arcs come to the whole way round. So each is the whole distance divided by however many there are, and several added together give any angle that is a fraction of a turn.

4:52Build the answer that way, from nothing but those two facts, and compare it with the formula at every whole angle in a turn: they agree three hundred and sixty times and disagree never. Angles that are not a fraction of a turn are caught between the fractions either side, and squeezing pins the arc to thirty decimal places. The formula was never a guess. It is the only rule that turning and adding leave available.

5:25Two quick numbers, using twenty-two sevenths for pi. A sixty-degree arc on a radius of three point five centimetres is a sixth of a circle whose way round is twenty-two, so the arc is three point six seven - a decimal that never ends, cut short. A hundred-and-twenty-degree arc on a radius of six point three metres comes out at thirteen point two exactly. Same formula, one answer exact and the other not.

5:58Now a trap. Ask for the perimeter of a slice - the pie-shaped piece between two radii. Radius fourteen centimetres, angle seventy-five degrees, and the arc is eighteen point three three. But the perimeter is the way round the outside, and two of its edges are straight. Add both radii, twenty-eight centimetres, and it is forty-six point three three - more than twice the curved part alone. Now the reason any of this matters on a Saturday morning.

6:34A running track is two straight sections joined by two curved ends, and those two ends together make one whole circle. Here are the measurements. Each straight is eighty-four point three nine metres. The inside edge of the innermost lane curves at a radius of thirty-six point five. Each lane is one point two two metres wide. And a runner does not hug the paint - they run about three tenths of a metre outside it, so their curve has a radius of thirty-six point eight.

7:11Add it up. Two straights come to a hundred and sixty-eight point seven eight metres. The two curved ends make one circle of radius thirty-six point eight: two hundred and thirty-one point two two metres. Together, four hundred point zero zero. That is where the four hundred comes from. But the lap is not exactly four hundred. It is four hundred metres and one point seven six millimetres. The straight and the radius were chosen to land it there.

7:45And the value used for pi matters: swap in twenty-two sevenths and the same lap comes out at four hundred point zero nine instead. The runner in lane two is one lane width further out, so their curve has a radius of thirty-eight point zero two. Their straights are identical to lane one's, because every runner runs the same straight. Their circle is bigger, and their lap comes to four hundred and seven point six seven.

8:18Seven point six seven metres longer. That is the stagger: lane two starts seven point six seven metres further along, and then both runners cover the same distance. The stagger is not a handicap. It is the cancellation of one. And between lanes two and three it is the same, and across every one of the seven gaps in eight lanes - identical, not merely close. Why identical? Subtract the two laps and watch what cancels.

8:54The straights go, because they are the same. What is left is two pi times the outer radius minus two pi times the inner one - two pi times the difference between them, and that difference is one lane width. The radius is gone. Not simplified away; it was never there. Sweep it across three hundred and ninety-one different bends, from a five-metre radius up to two hundred: one answer, every time.

9:25Do the same sweep on the area between the lanes and you get three hundred and ninety-one different answers - so it can see a dependence when there is one. So a two-hundred-metre track with the same lanes needs the same seven point six seven per lap. But there is a real difference, and it is not that one. On the four-hundred-metre track, fifty-eight parts in every hundred of a lap are bend; two-hundred-metre ovals with different straights give a hundred and sixty different answers.

10:01How much curve falls inside one leg of a relay does change with the track. What a lane costs you per lap does not.