The two properties that define a rectangle, and the one more a square needs

0:00Here is an arrangement: one rectangle, and four squares set out from its corners. Easy enough to point at which is which. So let me ask something that sounds silly. Is that rectangle in the middle a square? Obviously not. Now the other way. Are those four squares rectangles? Most people say no, quickly. Some people say yes, slowly. That disagreement is worth taking seriously. It happens because most of us learned these two words as pictures — this shape is a square, that longer one is a rectangle — and pictures cannot settle a question like that.

0:42But it is not a matter of opinion, and by the end of this you will be able to settle it in two lines. First we need to be able to say what we are testing, so here is one four-sided figure with everything named. Four corners. Label them A, B, C and D, going round. Four sides: A to B, B to C, C to D, and D back to A.

1:08And four angles, one at each corner, written with a little angle sign. That is the whole vocabulary. Corners, sides, angles. It is worth being fussy about, because the moment you start testing a figure you need to be able to say exactly which part failed. Everything that follows is a statement about the sides, or a statement about the angles, and nothing else. One more thing to fix before the tests: which sides count as opposite.

1:42Side A B runs along the top. The side facing it, across the figure, is C D at the bottom. That is one pair, and I will draw them in the same colour. The other pair is B C on the right and D A on the left. Second colour. So a four-sided figure has two pairs of opposite sides, and every side belongs to exactly one pair. Now watch what the tests actually ask about, because it is not always the same thing.

2:15Here is the test for a rectangle. It is two lines long. First: opposite sides are equal in length. The top matches the bottom, and the left matches the right. Notice it says nothing about the top matching the left. Second: all four angles are ninety degrees. Pass both and the figure is a rectangle. Fail either one and it is not. A shape can look perfectly rectangular and fail the second line by two degrees, and then it simply is not one.

2:51That is all a rectangle is. Not a shape you recognise. A figure that passes two checks. And here is the test for a square, also two lines. First: all four sides are equal. Not facing pairs. All four. Second: all four angles are ninety degrees. Read those two lines and then read the rectangle's two lines again, slowly. Something is going on between them, and it is the reason this whole video exists.

3:24So let me put them side by side. Rectangle on the left, square on the right. Look at the second lines first. All angles ninety degrees, and all angles ninety degrees. Those are not similar conditions. They are the same condition, written out twice. So the whole difference between the two tests sits in the first line. One says facing pairs must match. The other says all four must match.

3:55And all four matching is not a different demand from facing pairs matching. It is a heavier version of the same demand. Which means the square test is not a rival to the rectangle test. It is the rectangle test with the first line tightened. Which settles the question from the beginning. Take any figure that passes the square test. All four of its sides are equal. Then in particular its top equals its bottom, and its left equals its right, because they are all the same number.

4:33So it has already passed the rectangle's first line, without being asked. And its second line is the square's second line, word for word. So it passes that too. Every square is a rectangle. Not usually, not roughly. Every one, because passing the harder test includes passing the easier one. Think of it as one group sitting inside another: all the squares, drawn inside the much larger collection of rectangles.

5:07The other direction fails, and it fails in two different ways worth seeing. Here is a figure with four sides of exactly five centimetres each. Measure them, all five. So it passes the square's first line completely. But look at the corners. Not one of them is a right angle. It leans. Here is another one, four equal sides again, leaning further. Both would be called equal-sided by anybody who only measured sides. Neither is a square.

5:41So the second condition is doing real work. Equal sides on their own buy you nothing. And notice neither of those two is a rectangle either, because they fail the angle line as well. Now break the other one. This figure is tilted, so let me give you its sides as steps rather than as a picture. One side goes two across and one down. The next goes two back and four down.

6:11Those two directions are exactly perpendicular, and the same pattern repeats round the figure, so all four angles are ninety degrees. It passes the square's second line. And its opposite sides match, so it passes the whole rectangle test. But one side is twice the length of the other. All four are not equal. It fails the square's first line. Four right angles, and not a square. It is a rectangle, tilted.

6:44That leaves one question, and it is the good one. Could a four-sided figure have all four angles at ninety degrees, but opposite sides that do not match? It sounds like it should be possible. Nothing in the angle condition mentions lengths. So walk round the outline and see. Start along the first side, however long you like. At the corner, turn a quarter turn. Walk the second side. Turn another quarter turn.

7:15You are now facing exactly the way you came from. The third side runs back along the first. And to arrive home, it has to be the same length as the first, exactly. Anything else misses. So it cannot be done. The angle condition already forces the sides, which means one of the rectangle's two lines was never really free. We keep both anyway, because when you are checking a figure you have already drawn, measuring four sides is often quicker than measuring four angles.

7:52Time to use all of this. Here are four figures, and every one of them is tilted on purpose. Looking at them will not help you, which is the point. Their corners sit on grid dots, so you can count instead of measure. A: count the steps and all four sides come out the same, and all four corners are square. Both lines pass. A is a square. B: four equal sides, three across and four up each time. But the corners lean. First line yes, second line no.

8:27C: four equal sides again, leaning the other way. Same verdict. D: right angles all round, but the sides are five and ten. Second line yes, first line no. One figure out of four. And you found it by counting, not by looking. Two of them are not even rectangles. One of them is a rectangle and nothing more. And exactly one passes everything. One last thing, and it is what makes any of this usable.

9:00Read both tests again and notice what they never mention. The page. Neither condition says anything about being upright, or level, or square to the edge of the paper. So a square rotated forty-five degrees passes exactly the same two lines it passed before. It is not almost a square. It is a square, sitting at an angle, and the test does not care. That is what it means for a name to be earned by a test rather than by appearance.

9:33Two lines, and they decide it every time, on any figure, at any angle, whether or not it looks the part. Next time: building one of these from nothing, with a ruler and a set square, and checking it at the end.