The arc's length by the same share argument

0:00The wedge is settled. A slice opening theta degrees at the centre of a circle of radius r takes theta over three sixty of pi r squared. Now look at the edge of that slice. Not the two straight sides. The curved one. The arc. How long is it? The tempting answer is: same argument, different quantity, done. That is very nearly right, and the four lines are coming. But same argument is a claim, and the argument that settled the area ran on two facts, and neither of them was free.

0:40So write the argument out as machinery and see what it feeds on. First: turning changes nothing. Spin a wedge about the centre and what you measure does not move. Second: pieces laid end to end add. A wedge of thirty next to a wedge of forty is a wedge of seventy, and the measurements add exactly. Those two, plus one known value to hang the thing on, force the measurement to be proportional to the angle. Nothing else was used.

1:13So the question is not whether the argument transfers. It is whether those two facts hold for length round the rim. They look obvious. Obvious is where mistakes live. Take a bite of thirty degrees out of the rim, and measure the curve rather than the region. Now start the same bite somewhere else. Seven degrees round. Ninety. Two hundred and eleven. Twelve starting places in all, and every one of the twelve gives the same length.

1:46Turning is rigid. Nothing stretches and nothing shrinks, so a curve carried round the centre keeps the length it had. But that is not a fact about arcs. It is a fact about circles. Here is an oval, twice as wide as it is tall. Same bite, same twelve starting places. Only two of the twelve match. A bite at the end of the long axis is nearly twice as long as one at the end of the short axis. Here, where you start tells you how big the answer is.

2:22Property two. Two arcs, laid end to end. Measure from nought to sixty, measure from sixty to a hundred and twenty, and add. Compare that with the arc from nought to a hundred and twenty measured in one go. They agree, to the last place the measurement can see. Two pieces meeting at a point share only that point, and a point has no length. Now measure the wrong thing. The straight lines. The chords.

2:52The chord of a sixty degree opening is exactly one radius. Two of them end to end is two radii. The chord straight across a hundred and twenty is root three radii. About one point seven three. They do not add. The parts overshoot the whole by more than a quarter of a radius, in every case tried. Three objects, side by side. The arc of a circle: turning changes nothing, and pieces add. Twelve starts out of twelve, ten tests out of ten.

3:28The chord: turning changes nothing either, since a chord carried round the centre keeps its length. But pieces add in none of the ten. The arc of an oval: pieces add in all ten. But only two starts out of twelve survive turning. Each of the two is missing exactly one property, and neither is proportional to the angle. That is what the two properties are worth. Take away either one and the conclusion goes with it.

4:01So the argument survives. What it needs now is one number to hang on. One angle where the length is already known, without any of this. Take the whole turn. Three hundred and sixty degrees of arc is the entire rim, and the entire rim already has a value. It is the circumference. Two pi r. Roll the circle along a line for one revolution and the track it leaves is two pi r long.

4:31That is the anchor, and it is the only new thing in the derivation. Four lines, then. A whole turn, three hundred and sixty degrees, traces two pi r. One degree traces two pi r over three hundred and sixty. So theta degrees trace theta over three hundred and sixty, times two pi r. And that is the rule. Set it beside the one you already have. Whole turn: pi r squared. One degree: pi r squared over three sixty. Theta degrees: theta over three sixty, times pi r squared.

5:13Line for line, the same. Only the anchor changed. So if you have genuinely understood one of these, you have already done the work for the other. There is one derivation here, not two. Both answers out of one picture. A circle of radius twenty one, a wedge opening sixty degrees, and pi taken as twenty two sevenths. The whole rim is two, times twenty two sevenths, times twenty one. A hundred and thirty two.

5:45The arc is a sixth of that. Twenty two. The whole disc is twenty two sevenths times twenty one squared. One thousand three hundred and eighty six. The wedge is a sixth of that. Two hundred and thirty one. And sixty out of three hundred and sixty is a sixth. Arc over circumference. Wedge over disc. Angle over a full turn. Every one of them a sixth. That is not a coincidence to memorise. It is the derivation showing through.

6:20Which gives you the cheapest check there is. One of those rules makes a length and the other makes an area. Look at the power of r in each. Two pi r has one r in it. Pi r squared has two. So the arc rule carries r to the first power and the wedge rule carries it to the second. One answer is in centimetres, the other in centimetres squared.

6:46Doubling shows it plainly. Double r and the arc doubles. Double r and the wedge goes up four times. The two rules agree about the angle and disagree about the radius. That is the sharpest way to feel they are not one formula in two outfits. Four expressions. Exactly one is the area of a wedge opening p degrees in a circle of radius R. p over one eighty, times two pi R. One R. That is a length. Out.

7:22p over three sixty, times two pi R. One R again, and in fact that is the arc rule itself. Out. Two left, both areas. The powers have said everything they can say. So read the number underneath. p over one eighty, times pi R squared. Against the rule, p over three sixty times pi R squared, one eighty underneath makes it exactly twice too big. Out. p over seven twenty, times two pi R squared. The two on top and the seven twenty underneath cancel to three sixty, leaving p over three sixty times pi R squared. That is the answer.

8:09The power of r knocked out half the options in a second, and could not finish. Half a trick is worth having, so long as you know which half. Now a thing that catches people constantly. Back to that wedge. Radius twenty one, sixty degrees. What is the whole way round it? Not twenty two. The arc is one edge. The wedge has three, and the other two are radii, and each radius is twenty one.

8:41So the whole way round is twenty two, plus twenty one, plus twenty one. Sixty four. An arc is a piece of the curve. The boundary is everything you would walk tracing the edge with a pencil and coming back to where you started. Asked for the arc, twenty two. Asked for the perimeter, sixty four. Something the two rules say together that neither says alone. Take half the arc and multiply by the radius. Half of twenty two is eleven, and eleven times twenty one is two hundred and thirty one.

9:18That is the wedge. And it is not luck. Half of theta over three sixty times two pi r, all times r. The two and the half cancel, and one r joins the other to make r squared. Theta over three sixty, times pi r squared. So a wedge behaves like a triangle whose base is the arc and whose height is the radius. Measured directly rather than argued, half the arc times the radius lands inside the wedge on all twelve circles and angles tried. A third of it, a quarter, two thirds and the whole of it land inside none of them.

10:03Any rule with one unknown runs backwards. An arc of twenty two on a radius of twenty one gives back an angle of sixty. Halve the arc to eleven and the angle halves to thirty. Hand it the whole hundred and thirty two and it returns three hundred and sixty, which is the anchor arriving from the other side. Ask for an arc of two hundred on that circle and it refuses. No angle up to a full turn is that long.

10:35Or hold the angle and solve for the radius. An arc of twenty two opening sixty degrees needs a radius of twenty one; the same arc opening a hundred and twenty needs only ten and a half. And the most useful version: given nothing but a circumference of twenty two, read it as an equation. Two times twenty two sevenths, times r, is twenty two. So r is three and a half, and every other question is open.

11:09Last, a piece of wire. A brooch. A circle of silver wire thirty five millimetres across, with more wire forming five straight diameters through it. How much wire in all, and how big is one piece of the disc? First trap: thirty five is across, not out from the centre. The radius is seventeen and a half. The rim takes two, times twenty two sevenths, times seventeen and a half. A hundred and ten millimetres.

11:41The diameters take five lots of thirty five. A hundred and seventy five. Two hundred and eighty five millimetres altogether. One curved length and five straight ones, added without ceremony, because a millimetre is a millimetre whichever way it bends. Second trap: five diameters do not cut five pieces. Each diameter is two radii, so five of them cut ten. Ten wedges, three hundred and sixty over ten. Thirty six degrees each.

12:17Thirty six over three sixty, times twenty two sevenths, times seventeen and a half squared. Ninety six point two five square millimetres. One anchor, one fraction, and the whole of it read twice.