Why perimeter cannot stand in for area

0:00Every flat shape gives you two numbers. One is how far it is round the edge. The other is how much material is inside - how many unit squares it would take to fill it. And there is an obvious shortcut sitting here. Measuring round the edge is easy. You can do it with a piece of string. Counting the inside is slow work. So why not just measure the edge and be done with it?

0:29Before answering that, here is what a measure of material has to be able to do. Cut a shape into pieces. Whatever quantity we are measuring, the pieces have to total the whole. If they do not, the number is not measuring anything you could call an amount. The square count survives that. Cut the region anywhere and the counts of the parts add back up, because no material was created or destroyed.

0:59The boundary does not survive it at all. Cut a shape in two and you have made new edge - the cut itself, counted twice, once on each piece. The two boundaries total more than the one you started with. So they are already different kinds of number. But that alone does not tell you how badly the shortcut fails. Take two rectangles. One is seven across and four down; the other is eight across and three down.

1:31Go round the first: seven, four, seven, four. Twenty-two. Round the second: eight, three, eight, three. Twenty-two as well. Identical boundaries. Now count the insides. Twenty-eight squares against twenty-four. So the same boundary is sitting round two different amounts of material. Equal edge does not mean equal inside, and one pair of ordinary rectangles is enough to settle it forever. And these are not freaks. Among rectangles with sides up to twelve there are a hundred and twenty-five pairs that do exactly this.

2:12Fine, you might say. Equal does not force equal. But surely bigger forces bigger - a longer boundary must hold more. Here is one that is one across and twelve down. Round the edge: twenty-six. Inside: twelve squares. And here is one that is five by five. Round the edge: twenty. Inside: twenty-five squares. The first has the longer boundary, by six. And it holds less material, by thirteen. The two measurements are not merely failing to agree. They are pointing in opposite directions. Up to twelve by twelve there are two hundred and seventy-four pairs where that happens.

3:00Two examples kill the shortcut, but they do not explain it. For that, notice what we changed when we went from the square to the long thin one. We did not change the boundary by much. We did not set out to change the area at all. What we changed was how THIN the shape is. And thinness is a free choice. Fix the boundary and you have not fixed the shape - there is still a whole family of rectangles left, and you get to pick how stretched out yours is.

3:34That leftover choice is the whole story. Every one of the strange pairs comes from moving it, and neither of the two numbers has a name for it. So let us hold one measurement absolutely still and move the thinness on purpose. Twenty units of boundary. Five by five: twenty-five squares inside. Six by four: twenty-four. Eight by two: sixteen. Nine by one: nine. The boundary has not moved once. It was twenty at every step, because widening one side by exactly what you take off the other cannot change the total round the edge.

4:18But the inside has gone from twenty-five to nine, and it is still falling. Nine point nine by nought point one is still twenty round the edge, and holds nought point nine nine. Which raises a question worth asking properly. How small can it go? Name any amount of material you like. However small it is, there is a rectangle with a boundary of exactly twenty that holds less than that.

4:49You can always find one, because making the shape thinner shrinks the inside without ever touching the edge. There is no smallest member of this family. There is no floor. So a boundary of twenty tells you the area is at most twenty-five. It tells you nothing whatsoever about how much less. Now run it the other way. Fix the material and let the boundary do what it likes. Thirty-six squares. Six by six: twenty-four round the edge. Nine by four: twenty-six. Eighteen by two: forty. Thirty-six by one: seventy-four.

5:30Every one of those holds exactly the same amount of material. Not a square more or less. And the last one has more than three times the boundary of the first. Seventy two by a half is still thirty-six squares, and it is a hundred and forty-five round the edge. This direction is the more startling of the two, because now there is no ceiling. Name any length. A thousand. A million. There is a rectangle holding exactly thirty-six squares whose boundary is longer than that - make it that many units wide, and it will be.

6:11So for a fixed amount of material the boundary can be anything at all above a certain point, and for a fixed boundary the material can be anything at all below a certain point. That is what independent means. Knowing one of these numbers pins the other down on one side and leaves it completely free on the other. Here is a pair where you do not need any of that arithmetic.

6:40A strip twelve long and one wide: twelve squares of material, twenty-six round the edge. Now cut nine narrow slits into it, almost all the way through, like the teeth of a comb. The material barely changes - it drops from twelve to eleven point one nine. But the boundary goes from twenty-six to forty-two point two, because every slit adds two long new edges and takes almost nothing away.

7:11Twenty units of new boundary for every one unit of material given up. Slits are the cheapest way there is to buy edge. So put the comb beside a six by six square. The square holds more than three times the material and has little more than half the boundary - and you can see both of those without measuring anything. One more, and this one does not even change a sidelength.

7:40Take the five by four rectangle and lean it over into a parallelogram. All four sides are exactly as long as they were, so the boundary is still eighteen. It cannot be anything else. But the height has dropped, and the area with it. Twenty squares becomes sixteen. Lean it further and it becomes twelve. Lean it further still and the material keeps falling, with the same four sidelengths and the same eighteen round the edge the whole way down.

8:13Same four numbers on the sides. Any area you like. Now, independent does not mean unrelated. Look back at the family with a fixed boundary: the biggest inside belonged to the square. And the family with fixed material: the shortest boundary belonged to the square as well. That is a real relationship. But look at its shape. It says a boundary of twenty holds at most twenty-five squares, and thirty-six squares needs at least twenty-four of boundary.

8:49At most. At least. It is a limit in one direction, and it names the extreme shape - not a formula, and never an answer. Which is why the material gets counted rather than measured round. The edge of a shape is a genuine thing to know. It is simply not a way of finding out what is inside.