Same area, many perimeters: why one does not determine the other

0:00You now have two ways to measure a shape. How far round the edge, and how much is inside. And almost everybody treats those as one measurement wearing two hats. Bigger shape, bigger both. That feeling is wrong, and this section of the book exists to take it away from you. So picture two dials. One reads the area, one reads the perimeter. The claim we will establish: you can hold either dial still and swing the other one wildly.

0:34Start with the tangram, and a question the book slips in at the end of that set. Take the seven pieces and build the square. Then take them apart and build a rectangle. Now: is the distance round the rectangle the same as the distance round the square? Notice you are already sure about the area. Nothing was added, removed or cut, so it is identical — no arithmetic needed.

1:02The perimeter is a separate question. And the answer is no, the boundaries do not match. Which is the whole topic, before any numbers turn up. Same pieces. Same area. Different edge. Now make that precise, with the exercise the book sets next. Draw every rectangle with whole-number sides and an area of twenty-four square units. That is a factor-pair question. One and twenty-four. Two and twelve. Three and eight. Four and six.

1:37Four rectangles. Every single one covers exactly twenty-four square units. Now work out the perimeter of each. One by twenty-four: two lots of twenty-five — fifty. Two by twelve: two lots of fourteen — twenty-eight. Three by eight: twenty-two. Four by six: twenty. Fifty, twenty-eight, twenty-two, twenty. Same area, every time. And the boundary more than doubles across the list. Put the two ends of that list side by side, because the picture explains it.

2:16The one by twenty-four is a long thin ribbon. The four by six is a compact block. Both hold twenty-four squares. One has a boundary of fifty, the other of twenty. And you can see why, just from where the squares touch. In the ribbon, almost every square has two exposed sides. Hardly anything is hidden inside. In the block, the middle squares are surrounded — their edges are shared, and shared edges are not boundary.

2:48That is the whole mechanism. Perimeter measures exposure, not amount. Now the book asks the real question. Given any area, can you say which rectangle has the biggest perimeter, and which the smallest, before listing them? It prints no answer, and neither does the key. So here is mine. The greatest is always the one by the area itself — the longest, thinnest strip the factors allow. The least is always the factor pair closest together, because that is the most compact rectangle you can build.

3:27Test it on thirty-two, which is the book's next one. Factor pairs: one and thirty-two, two and sixteen, four and eight. Perimeters: sixty-six, thirty-six, twenty-four. Biggest is the strip. Smallest is four by eight, the closest pair. And notice what this means. To get a small boundary you want the sides as equal as possible — you want it as square as it can be. Now the book drops rectangles and gets more interesting. Nine unit squares, and any shape you like.

4:05Three rules. Every square meets another along a whole side. One piece. No holes. Here is the obvious build: three by three. Perimeter twelve. And one the book prints beside it: four squares in a top row, four in a bottom row, one joining them at the left end. Perimeter twenty. Twelve and twenty. Same nine squares — you could pick them up and rearrange one into the other. The area dial has not moved at all. The perimeter dial has gone from twelve to twenty.

4:42The book then asks what the smallest and largest possible perimeters are, and whether more than one shape achieves each. I did not want to guess at that, so I had a computer build every single shape nine squares can make under those three rules. There are one thousand two hundred and forty-eight of them. And the answer is sharper than I expected. The perimeter is always twelve, fourteen, sixteen, eighteen or twenty. Never anything else. Never odd, never below twelve, never above twenty.

5:19Twelve is reached by exactly one shape — the three-by-three block, and nothing else. Twenty is reached by eight hundred and fifteen different shapes. So the extremes are not alike. The most compact shape is unique; the most stretched-out one is thoroughly ordinary. And eighteen, which the book asks you to build, has three hundred and thirty-four shapes. You will find one. Now the sharpest thing on the page. A figure of eleven unit squares, perimeter twenty-four.

5:55And one more square, waiting to be attached. The question is what the perimeter becomes — without recounting the whole boundary. Stick the square on so it touches along one edge. The perimeter goes up by two. Twenty-six. Now put the same square into a step, so it touches along two edges. The perimeter does not change at all. Still twenty-four. Now tuck it into a notch with squares on three sides. The perimeter goes down by two. Twenty-two.

6:28Same square. Same added area, every time. And the boundary went up, stayed put, and came down. Here is why, and it is one line. The square you brought in has four edges. Each edge that ends up touching the figure hides two edges — its own, and the one it landed against. So the change in perimeter is four, minus two for every edge you hid. One shared edge: four minus two is plus two. Two shared: four minus four is zero. Three shared: four minus six is minus two.

7:08And now look hard at that line, because of what is not in it. The area of the square you added never appears. Not once. The perimeter change depends only on how the piece was joined — never on how much you brought. That is the mechanism behind this whole video. Area cares how much. Perimeter cares how it is arranged. Now the book puts it into a house, which makes it matter.

7:38Two rectangular plots, with the room measurements printed on the plans. On the first: bedrooms take a fifteen-foot column, the toilet a five-foot strip, the kitchen a fifteen-foot block. That is thirty-five feet across, thirty down. Thirty-five by thirty. One thousand and fifty square feet. On the second: the bedroom column is twelve feet wide, fifteen plus ten deep. So twenty-five deep, forty-two across. Forty-two by twenty-five. One thousand and fifty square feet. Exactly the same.

8:17But go round them. One boundary is a hundred and thirty feet. The other is a hundred and thirty-four. Identical land, four extra feet of wall. Walls are bought by the foot, and someone pays for those four. And now the question that proves you have got it, because you build the counter-example yourself. Draw two shapes. The first covers eighteen square units, the second twenty — and the first must have the longer boundary.

8:51If the two marched together that would be impossible. But it is easy. Make the eighteen a single-file strip, one by eighteen. Its boundary is thirty-eight. Make the twenty a compact four by five block. Its boundary is eighteen. Eighteen square units with a boundary of thirty-eight, against twenty square units with a boundary of eighteen. Less area, more than twice the edge. One more, which catches people. Fold a square in half and cut along the fold.

9:25Each piece has exactly half the area — that part behaves. But each piece's boundary is three-quarters of the original square's, not half. And the two pieces together have half again as much edge as the square you started with. Cutting made no new area and a great deal of new boundary. The chapter summary states both halves of this outright, and it is worth reading slowly. Figures with equal areas can have different perimeters.

9:59And figures with equal perimeters can have different areas. We proved the first half several times over — the tangram, four rectangles of twenty-four, nine squares between twelve and twenty, two houses on the same land. And you met the second half a while back without noticing. A perimeter of twenty gives one by nine, two by eight, three by seven, four by six, five by five — areas from nine up to twenty-five.

10:32So neither measurement fixes the other, in either direction. Two genuinely different questions about a shape — and from now on, expect them to answer independently. Next time, the shape that finally lets us pin an area down without counting a single square: the triangle.