Perimeter as a walk around the border, and why perimeter-to-side ratios are fixed

0:00Look down a curved running track and something is odd. The runners are not lined up. The marks climb outward in a staircase, each lane starting a little ahead of the one inside it. That offset has a name: the stagger. So, two questions. Does the stagger favour the runner on the inside or the one on the outside? And how would anyone work out how big it should be?

0:27Both are answerable, and neither is answerable yet. What is missing is a way to measure the distance round a bend, and that turns out to need an idea worth the whole of this video. Start with the word. A perimeter is not a formula. It is a walk. Put an insect on the border of a shape and let it walk all the way round, back to where it began, without ever doubling back.

0:57The distance it covers is the perimeter. That description has one enormous advantage over a formula. It works on a border of any shape whatsoever. A shape with nine sides, a shape with one straight edge and one curved one, a shape nobody has a name for. There is no formula for those, and there does not need to be. Two things about that walk are worth pinning down, because everything later leans on them.

1:29The first: you can cut it anywhere. Put a mark halfway along one side and walk it as two pieces instead of one, and the total is exactly the same. Which is obvious, and is also the entire reason a border with no formula can still have a length. The second: you must not wander. Step off the border and back on again and the walk gets longer. Not sometimes longer.

1:58Strictly longer, every single time - checked over a thousand and twenty detours, and not one of them came out equal or shorter. The straight run between two points is the short one, and that fact will come back at the end. Now the shapes you do have formulas for - and the claim that you never needed to remember them. Take a square with side a. Walk it. One side, two sides, three, four, each of them a.

2:29Four a. That is not a formula recalled; it is four edges counted. Do the same on a triangle with all three sides equal and you count three of them: three a. Now a rectangle, length a and width b. Two of the edges are a and two are b, so the walk comes to two lots of a plus b. The number in front is always just how many equal edges there were.

2:59Look at those last two again. Twice a plus b, for the rectangle. Four a, for the square. They are the same statement. Take the rectangle and impose one extra condition - make the width equal to the length - and twice a plus a is four a. The square's formula was inside the rectangle's the whole time, waiting for that one condition. This is a habit worth picking up early.

3:28When two results look different, the interesting question is usually whether one of them is the other with something extra demanded of it. Here is where it stops being bookkeeping. Take a square of side two. Its perimeter is eight. Compare the perimeter to the side: eight to two, which is four to one. Now a square of side four. Perimeter sixteen; sixteen to four is four to one again.

3:59You might reasonably suspect the doubling is doing the work, so try one that is not a double of anything: side two and a half. Perimeter ten. Ten to two and a half is four to one. Three squares is evidence. It is not a reason, and a reason is what we are after. Here is the reason, and it is one sentence long. When you enlarge a shape, ONE factor multiplies every single edge.

4:30Not a different factor for each side. The same one, everywhere. So the perimeter, which is those edges added up, gets multiplied by that factor too - and so does the side you are comparing it to. Top and bottom of the ratio grow together, and a ratio whose two halves are multiplied by the same number is the ratio you started with. It cannot move. That was tested on two thousand one hundred combinations of shape and scale factor, shrinking as well as growing, and the ratio held every time.

5:09And it holds for ratios that have nothing to do with perimeters either - one side against another side - because the argument was never about perimeters. Be careful what you take from that, though. Four to one is not something four sides buy you. Rectangles have four sides. Compare a rectangle's perimeter to its longest side and, across the ordinary whole-number rectangles, that comparison takes twenty one different values.

5:42They run from nine quarters up to fifteen quarters, and not one of them is four. Four to one is bought by four EQUAL sides. And that includes shapes that are not squares at all - a leaning diamond has four equal sides and gives four to one just as happily. The number belongs to a shape, in the strict sense: to a figure and every scaled copy of it, and to nothing else.

6:11Which brings us to the shape all of this has been heading towards. A circle. Every argument so far has compared the perimeter to a side. A circle has no side. It has no edges at all to count, so there is no number to put in front of anything, and nothing obvious to put underneath the comparison. Put a square, an equal-sided triangle and a circle in a row and mark the length each one is measured against.

6:43Under the first two, a side. Under the third, a blank. So what goes in the blank? The circle does have lengths of its own - the radius out from the centre, and the diameter straight across. Take the diameter. And now notice that the argument from before does not care in the slightest that there are no sides to count. Enlarge a circle and one factor multiplies everything in it: the diameter, the radius, and the distance round the outside.

7:18Top and bottom again. So the perimeter of a circle, compared to its diameter, is one fixed number - the same for a coin, a wheel and a running track. And we can already put a foot on it. A six-cornered figure with its corners on the circle has every side exactly equal to the radius - six radii, which is exactly three diameters - and every one of its sides cuts across the inside.

7:48By the straight-line fact from earlier, the circle's own border must be longer than that. More than three diameters, and we have not approximated anything. What is a fixed ratio actually for? This. Two circles, and all you are told is that the distances round them are in the ratio five to four. What is the ratio of their radii? Five to four - and you can say so without knowing the constant at all, because the same constant stands on both sides and cancels.

8:24Change it to anything you like and the answer does not budge. Compare that with a question that does need it: a circle whose border measures forty four, find the radius. For that you must put a value in, and the usual instruction is to take the constant as twenty two sevenths, which gives exactly seven. One warning while we are here. Compare the border to the DIAMETER and you get one constant; compare it to the RADIUS and you get another, exactly twice as big.

8:59Being vague about which is the single most reliable way to lose a factor of two in all of this. So, back to the runners. Can we settle the stagger yet? No - and that is the honest answer rather than a dodge. What the staggered start is doing is removing a difference, not creating one, but to say by how much you need the distance round a bend, and that is a number this video has bracketed rather than found.

9:30What we do have is the shape of the answer. One measurement, and a ratio that cannot move, and the whole boundary follows. That is what a fixed ratio buys: you never have to measure the border of anything round again. You measure one straight line across it, and the rest is already decided.