A sector's area as its share of the full turn

0:00Start with the one thing you already have. A filled circle of radius r measures pi r squared. That is not being derived here. It is the fact everything else in this video is built out of, so write it under the disc and leave it there. Now notice something about that disc that is easy to walk straight past. It is already a wedge. Two radii bound it, the same radius drawn twice, once as the start and once as the finish, and the angle they open at the centre is a full turn.

0:36Three hundred and sixty degrees, and pi r squared. So here is the table we are trying to fill in. On the left an angle. On the right the area of the wedge that angle cuts out. One row is already written: a full turn takes the whole of pi r squared. What we want is the row for any angle at all. And the move everybody makes here, the move that feels like no move at all, is to say that a wedge of thirty degrees takes thirty three hundred and sixtieths of it.

1:12Half the angle, half the area. That is not a definition, and the picture does not force it on you. It is a claim, and a claim can be wrong. Why be suspicious of something that obvious? Because the very same sentence is false one step away. Take the same two points on the rim, and cut with the straight line between them instead. The chord. Keep the piece past it.

1:41Now double the angle. The piece does not double. It much more than doubles. So being proportional to the angle is not a property of circles, or of pieces of circles. It is a property of this particular piece, and something has to be true of the wedge that is not true of its neighbour. Two things are, and both are worth naming out loud. The first is that turning changes nothing.

2:11Take a wedge of thirty degrees and turn it about the centre. Its area does not move. Not approximately. Turning is rigid: no stretching, no squashing, nothing gets longer or shorter, and area is exactly the thing a rigid motion leaves alone. So a wedge of thirty degrees starting here, and a wedge of thirty degrees starting anywhere else, are the same size. Equal angles cut equal areas. Where a wedge sits is not information about how big it is.

2:48The second is that wedges laid side by side add. Put one wedge against another along a shared radius. The angles add, plainly. Twenty and forty make sixty. And the areas add too. The two pieces have nothing in common but the radius between them, and that radius is a line: it has no thickness and no area. They overlap in nothing and they leave nothing over. Angles add, areas add, in step.

3:20That is the second property, and together with the first it is very nearly enough. Very nearly. And to see how much work those two are doing, watch the second one break. Back to the piece past the chord. Take the cut from here to sixty degrees, and then the cut from sixty to a hundred and twenty. Each one leaves a thin sliver past its own chord. Now make the single cut from here all the way round to a hundred and twenty.

3:52That piece is not the two slivers put together. It is the two slivers plus a great triangle sitting between them, and the triangle is most of it. These pieces do not tile. They leave something over. Held against the whole disc, the two of them fall short of the one by forty nine degrees worth of disc. That is the difference between a shape that gets a rule and a shape that does not.

4:21So, the derivation. It is three lines. Three hundred and sixty degrees carries pi r squared. Cut the disc into three hundred and sixty wedges of one degree each. By the first property they are all the same size, and by the second their areas add up to the whole. So one degree carries a three hundred and sixtieth of pi r squared. And an angle of theta degrees is theta of those one degree wedges laid side by side.

4:55So theta degrees carries theta over three hundred and sixty of pi r squared. The middle line is the only one with anything in it. The other two are bookkeeping. There is one gap left in that, and it is worth seeing rather than stepping over. Those two properties settle every angle that is a whole number of degrees, and every angle that is any fraction of a turn you can write down.

5:24What they do not settle, on their own, is an angle that is no fraction of a turn at all. Here is a rule that passes both tests and is still wrong. It depends only on the sweep and not on where the sweep starts. It adds when angles add. It gives the whole disc for the whole turn. And it hands nothing whatever to this wedge, which sits between a hundred and twenty seven degrees and a hundred and twenty eight, and which is plainly more than a third of the disc.

5:58What rules that out is a third thing area does, so ordinary it is easy to forget: a piece inside another piece cannot measure more than it. Squeeze the awkward angle between the whole degrees on either side of it, and it has nowhere left to go. So here is the rule, and here is what each symbol has to be. The area is theta over three hundred and sixty, times pi r squared. Two inputs, and nothing else.

6:30r is a length, so it arrives in centimetres or metres, and it appears squared, which is why the answer comes out in square centimetres or square metres. Theta is an angle, and the three hundred and sixty underneath it is there for exactly one reason: because theta is being counted in degrees. Count the angle some other way and that number changes with it. The three hundred and sixty is not decoration. It is the units.

7:04One worked all the way through. A disc of radius four, a wedge of thirty degrees, and pi taken as three point one four. The whole disc first: three point one four times sixteen is fifty point two four. Now the share. Thirty over three hundred and sixty. Do not reach for a calculator. Cancel it. Thirty goes into three hundred and sixty twelve times, so the share is one twelfth.

7:35Fifty point two four over twelve is four point one eight six and so on, which we report as four point one nine square centimetres. Cancel first, multiply second, and round once, at the end. The other piece, the big one, you can reach two ways, and it is worth doing both. Subtract. Fifty point two four take away four point one nine is forty six point zero five. Or run the same rule on the angle that is left over.

8:10Three hundred and sixty less thirty is three hundred and thirty, and three hundred and thirty over three hundred and sixty of fifty point two four is forty six point zero five as well. The same number, to the last figure. That agreement is not luck and it is not a coincidence. It is the adding property from earlier turning up in the arithmetic: if the two pieces really do come to the whole, then the two routes cannot disagree.

8:43Which makes it the cheapest check you will ever get on the claim the whole rule rests on. Most questions do not hand you an angle. They hand you a description, and the whole job is getting to the angle first. A minute hand sweeping five minutes. Five minutes is a twelfth of the dial, and a twelfth of three hundred and sixty is thirty degrees. Not five. An umbrella with eight ribs, evenly spaced. Eight equal gaps, so three hundred and sixty over eight, forty five degrees each.

9:20A goat tethered at the corner of a square field. The corner is a right angle, so it grazes a quarter of a disc, and with a rope of five metres that is nineteen point six two five square metres. Lengthen the rope to ten and it becomes seventy eight point five. Four times as much grass, from twice the rope. And check that the rope still fits: ten metres against a fifteen metre side, so the quarter disc is still inside the field.

9:55Two windscreen wipers, each blade twenty five long, each sweeping a hundred and fifteen degrees. The warning you will hear is: do not put two hundred and thirty into the rule. Put a hundred and fifteen in and double the answer. But look at what those two actually give you. The same number. They have to. The rule is linear in the angle, and being linear in the angle is exactly what proportional means.

10:24The real reason to sweep one blade and then double is that the two blades cover different ground, and the arithmetic is not what tells you so. Push the idea and it does break properly. If each blade swept two hundred degrees, doubling the angle would ask for four hundred, and four hundred degrees is not an angle you can cut a wedge at. Two blades is two regions. It is never one bigger one.

10:55Last, a question that looks like arithmetic and is not. Four expressions are offered for the wedge cut by an angle p from a circle of radius capital R. p over one eighty, times two pi R. p over one eighty, times pi R squared. p over three sixty, times two pi R. And p over seven twenty, times two pi R squared. Compute nothing. Look at the powers of R.

11:27Two of them carry R to the first power, and R to the first power is a length. Those two cannot be areas at all. One of them is the length of the arc, and the other is twice the length of the arc. That leaves two, and now the divisor decides between them. p over seven twenty times two pi R squared is p over three sixty times pi R squared, which is the rule.

11:56Check the units before you check the arithmetic, and three quarters of the work is done before you have written anything down. One measured fact, and one claim about angles that had to be earned. That is the whole of it.