Estimating the area of a shape with no formula

0:00Two shapes. One a smooth wandering outline, the other spiky, with about nine points. Which one encloses more? Have a guess — an actual guess, before you read on. Most people say the spiky one is smaller. All those points make it look thin. I am not going to tell you the answer, and neither does the book. The guess is not there to be marked. It is there to make you want a method.

0:28Because right now you have no way of settling it, and that is the uncomfortable part. So why can you not just work them out? Every area rule you have met needs the shape to repeat something. A rectangle has equal rows. A square has equal everything. These two have nothing to repeat. No equal rows, no equal sides, no formula, and there never will be one. But look at them. They obviously enclose some definite amount of space.

1:02So the area exists. What is missing is a way to get at it. And the method the book gives you works on absolutely any shape — which is a much bigger prize than a formula. Here is the method. Lay transparent paper over the shape and trace the outline. Now lift the tracing and lay it on squared paper. That step matters more than it looks. The shape did not start on a grid — you brought the grid to it.

1:33Which means this works on a leaf, a footprint, a country on a map, anything you can trace. Now count the squares underneath. And that is where it gets interesting, because most of them are not whole. The book gives four rules for counting. Here is the first, and it is the easy one. One whole small square of the sheet counts as one square unit. Nothing to argue with. It is entirely inside the shape, so it is entirely counted.

2:06Deep inside the outline there will be lots of these, and they cost you no thought at all. The whole difficulty is at the edge — the squares the boundary cuts through. Three rules left, and all three are about those. Rule two. If less than half of a square is inside the shape, ignore it. Count it as nothing. Rule three. If more than half is inside, count the whole square. Count it as one.

2:36Now, both of those are wrong, and it is worth being blunt about that. Rule two throws away a sliver of area that really is inside the shape. It under-counts. Rule three claims a corner that is outside the shape. It over-counts. So a fair question is why anyone would accept a method built on two deliberate mistakes. Here is the answer, and it is the best idea in the section.

3:06The two mistakes point in opposite directions. Walk round the boundary of a real shape and you meet both kinds of square, over and over, roughly turn about. Every square you gave away is paid back by a square you took. They do not cancel exactly — nobody claims they do. But they cancel largely, and they keep cancelling all the way round. That is why this is an estimate and not a guess. A guess has no error you can reason about. This one does.

3:40And it tells you exactly how to do better. Use a finer grid. Halve the spacing and the boundary squares get smaller, so the amount in dispute shrinks — while the true area does not move at all. Rule four. If the boundary cuts a square exactly in half, count one half. And notice that rule four is not like the other two. It is not a compromise. It is exactly right.

4:09Half a square is inside, so you count half a square. No error is introduced at all. Which is a small hint about something bigger. When the boundary happens to land somewhere the grid can describe precisely, the counting stops being approximate. Hold on to that, because it is about to stop being a hint. Here are four figures drawn on a grid of dots, with every corner sitting on a dot.

4:39Now nothing is in dispute. Every boundary square is either whole, or cut cleanly by an edge running corner to corner. So count them: whole squares, plus halves, and no judgement calls anywhere. The first comes to four square units. The second, nine. The third, ten. The fourth, eleven. And those are not estimates. They are exact. Same counting, same four rules — but with the boundary following the grid, rule four does all the work, and rules two and three never come up.

5:18Now something worth doing with those same four figures, because the book measured them twice. Earlier in the chapter it went round their edges instead, and wrote the perimeters in straight and diagonal steps. Eight s plus two d. Four s plus six d. Twelve s plus six d. Eighteen s plus six d. So put both numbers under one figure. Area four. Perimeter eight s plus two d. Two measurements of one drawing, in two different units, answering two different questions.

5:55And look at the second and third figures. The second has area nine, the third has ten — so the third is bigger. But round the edge, the second is four s plus six d and the third is twelve s plus six d. The third is very much longer. The two numbers are not locked together. Bigger round the edge does not mean bigger inside, and we will push on that hard next time.

6:25Now go and read the chapter summary, because it says something very carefully. It says a region can be estimated, or determined exactly, by breaking it into unit squares. Estimated, or determined exactly. Both, in one sentence, and it is not hedging. One procedure covers two cases, and which case you are in depends entirely on the shape. If the boundary follows the grid, you get exactness. If it wanders across squares, you get an estimate.

7:00Same rules, same counting, either way. That is a remarkably economical piece of mathematics. And two tasks the book sets, which are the whole point of learning this. First: find the area of the floor outside your corridor, in square metres. Second: find the area of your school playground, in square metres. No formula is going to help you. Real floors have pillars and alcoves; real playgrounds have corners nobody squared off.

7:35So pace it, sketch it on squared paper, and count. Whole squares, ignore the slivers, take the big ones, halve the exact halves. You will get a number. It will be an estimate, and you will know roughly how good it is, and you will know how to make it better. That is the difference between measuring something and guessing at it. So look back at what these four small rules actually bought you.

8:04Before them, you could only find the area of shapes that repeat — rectangles, squares, and that is nearly the whole list. After them, you can find the area of anything you can draw. The price was exactness at the boundary, and the book is honest that you paid it. But it also arranged the errors to point opposite ways, so the price stays small — and told you how to make it smaller.

8:32And on the shapes where the boundary behaves, you pay nothing at all. A rule that always works and is sometimes approximate beats a rule that is always exact and almost never applies. Next time, the question those four figures raised: how far round and how much inside are genuinely independent — and just how far apart they can be pulled.