Constructing a square or rectangle from its side lengths

0:00Draw a square with sides of six centimetres. It sounds like an instruction you could follow without thinking about it. But look at what you have actually been given. One number, and one word. Six centimetres, and square. Everything else is yours to decide: where the figure sits, which way up it is, where the pencil goes first. And the tempting first move is the wrong one. Reach for the ruler, measure out four sides of six, and you can easily end up with something that is not a square at all.

0:36Four equal sides with nothing controlling the corners is a figure that can lean over as far as you like, and it still has four equal sides. So start with the one thing the instruction hands you outright. A side, six centimetres long. Put it along the bottom, and letter its two ends P and Q. That is the first step, and it is the whole of the first step. One line, measured once.

1:06Now notice what has already been used up. The number six has been spent on this side. The word square has not been touched yet. Everything from here on is that word. The word square is two conditions. All four sides the same length, and all four angles ninety degrees. The second one is what tells you where the next line goes. At P, the corner between this side and the next one has to be a right angle.

1:37So the next line has to leave P at ninety degrees to PQ. There is no other option available. That is not a stylistic choice, and it is not because squares happen to look like that. It is the condition, turned into an instruction for your hand. Set the right angle at P, and draw the line rising from it. Now, how far up that line does the next corner sit?

2:05Six centimetres. Because the other condition says all four sides match, and PQ is six. So lay the ruler along the new line, find six, and mark it. Call that point S. The side condition is doing all the work at this step. The angle condition already did its job when the line was drawn. Three corners placed now, counting the two you started with. And both of the square's conditions have now been used exactly once each.

2:39There is a second way to make that same mark, and it is worth the detour. Instead of the ruler, open a compass so its point sits on P and its pencil sits on Q. The compass is now holding the length of PQ. Not a number — the actual length. Swing it up to the new line, and mark where it crosses. Same point. But not the same reliability, and this is the part worth remembering.

3:08If your first measurement was slightly off, the ruler asks you to read the scale a second time, and two slightly different readings give you two sides that differ. The figure stops being a square. The compass gives you two sides equal to each other whatever that first reading was. Wrong size, perhaps. Still a square. Now the other bottom corner. At Q, the angle between PQ and the side going up also has to be ninety degrees, for exactly the reason it did at P.

3:45So set the right angle at Q, and draw a second line rising. Two verticals now, one at each end of the base. And here the construction quietly changes character. Up to this point, every step has added something you chose. From here on, the figure starts finishing itself. The compass is still open at six centimetres. Nobody has touched it. Put its point on Q, swing it across, and mark where it meets the second vertical.

4:18That is the fourth corner. Call it R. Join S to R, and the figure is closed. Now look back at what just happened, because it is easy to miss. Nothing was measured. The compass was not reset, the ruler was not picked up again. No new number entered the drawing at all. So here is a fair question to ask of a finished figure. How long is that last side, the one running from S across to R?

4:49And what are the angles at S, and at R? You never set either of those angles, and you never measured that side. You could go and check: six centimetres, ninety degrees, ninety degrees. But checking is not the same as discovering. All three of those answers were decided the moment the first three corners went down. Which is worth taking seriously, because a construction is easy to read as six lucky steps that happen to land on a square.

5:24Here is why there was no luck in it anywhere. With P, Q and S on the paper, ask where R could possibly go. The angle at Q has to be a right angle, so R has to sit somewhere along the vertical through Q. The angle at S has to be a right angle too, so R has to sit somewhere along the horizontal through S. Two lines that are not parallel cross at exactly one point.

5:53R has nowhere else to be. That is what it means to say the fourth corner is determined. Not that it is likely, or that it usually works out. That it has no room left to be anything else. And none of this was special to squares. Give any four-cornered figure four right angles, and its opposite sides are forced to be equal. There is no figure anywhere that manages one without the other.

6:24Now change the numbers, and watch how little else changes. Four centimetres and six centimetres, and this time you want a rectangle. Draw the four centimetre side. Right angle at each end. Mark six up one of the verticals, carry that same six across to the other with the compass, and close it. The same four decisions, in the same order. Try two centimetres by ten, which looks like a much more awkward drawing and is not.

6:56Long and thin, certainly. But every single step is the step you have just done twice. And notice where the square went. A rectangle turns out to be a square exactly when those two readings are the same number. The square was never a different construction. It was this one, with the second reading copied from the first. One more, because it shows the method being used rather than practised.

7:26Take a rectangle eight centimetres wide and four tall, and fit a square inside it so the two share a centre. How big is that square? If it is to reach both of the long sides, its height has to be four. So its side is four. And because the two centres agree, whatever is left over has to be split evenly between the two ends. Eight take away four leaves four, so that is two centimetres at each end.

7:56Both of those numbers came out of the conditions. Neither of them was measured off a drawing, which is just as well, because the drawing is the thing you were trying to produce. So look again at what those six steps really were. Two of them were the angle condition, applied at the two corners where you still had a choice. Two were the side condition: once to set a length, and once to carry it somewhere else without re-reading it.

8:27And the last two were the figure closing itself, because by then there was nothing left to decide. A construction is not a recipe you follow and then hope. It is a list of conditions, put in the order your hand can reach them. Which is exactly why it can end with a check, instead of a wish. Measure the last side, measure the two angles you never set, and if they are what the conditions said they would be, the figure is right — and you knew it before you measured.