Two independent ways to classify a triangle: by side and by angle

0:00Here is a pile of triangles, no two of them the same shape. You could sort them. But into what? There are two ways, and they are not two halves of one list. You can sort a triangle by its sides. Or you can sort it by its angles. Two separate questions, put to the same triangle. And every triangle answers both. It comes away with one label from each.

0:27Which means a triangle is never just one kind of thing. Sides first. Look at the three sides and ask one simple question. How many of them are the same length? All three the same. That one has a name: equilateral. Exactly two. Isosceles. None at all, three different lengths. Scalene. Three, two, or none. There is no fourth answer, so every triangle lands in one of those three. And notice what that test looked at. Lengths, and nothing else.

1:05Nobody has measured an angle yet. Which is the whole point of what follows, and we will come back to it. Now, that first sorting raises a practical problem on paper. A drawing is never accurate enough to be trusted. Two sides meant to be equal will not quite look it. So there is a notation. A small stroke drawn across a side. Two sides carrying the same stroke are being declared equal.

1:36One stroke on each of the three sides. Equilateral, on purpose, not by accident of drawing. Strokes on two sides only. Isosceles, and the third side is free to be whatever it likes. No strokes anywhere at all. Scalene. Those strokes are not decoration. Take them away and the first two pictures are the same picture. Before the second question, it is worth knowing how common each of those three actually is.

2:09So I took every triangle you can build with whole-number sides, none of them longer than sixty centimetres. Nineteen thousand three hundred and seventy-five of them. Sixty are equilateral. Two thousand six hundred and forty have exactly two sides the same. All the rest, sixteen thousand six hundred and seventy-five, are scalene. Eighty-six per cent. So scalene is not the odd one out. It is what a triangle is, unless something has been done to make it otherwise.

2:46One more thing about isosceles. The condition is two equal sides. It does not say exactly two. So whether an equilateral triangle also counts as isosceles is a line drawn differently in different places, and I am not going to draw it here. Second question, and the same triangle. Ignore the sides completely and look at the angles. One of them is a right angle. That is a right-angled triangle.

3:19Perfectly familiar. It is the one drawn with a little square in the corner, and that square is a claim, not a decoration. Next. One angle bigger than a right angle. An obtuse-angled triangle. And notice the word: one. Exactly one. It cannot have two, and the reason takes a second. The smallest pair of obtuse angles is ninety-one and ninety-one. That is a hundred and eighty-two already, and there is still a third corner to pay for.

3:53So an obtuse-angled triangle carries exactly one obtuse angle. The same argument rules out two right angles. Which leaves the third name, and this is where it gets interesting. The obvious guess: an acute-angled triangle is one that has an acute angle. It sounds right, it reads right, and it is completely useless as a definition. Here are three triangles that could hardly look less alike. One has a right angle. One has an angle of a hundred and thirty. One has nothing bigger than seventy.

4:33Every single one of them has an acute angle. In fact every one of them has two. And a class that everything belongs to has not sorted anything at all. And that is not a fluke of the three examples I happened to pick. Suppose a triangle had two angles that were each ninety or more. Together, those two would already be a hundred and eighty. The whole budget. Nothing left for the third corner, and a corner of nothing is not a corner.

5:07So no triangle manages it. At most one angle reaches ninety, which leaves at least two below it. Every triangle whose angles are whole numbers of degrees is two thousand seven hundred different shapes, and I checked all of them. Not one has fewer than two acute angles. Which is exactly why the definition has to say all three. An acute-angled triangle is one whose three angles are all acute.

5:38All three under ninety. Not one of them, not two. Three. And that definition does what a definition is for. It leaves things out. Of those two thousand seven hundred shapes, six hundred and seventy-five have all three angles acute. Forty-five of them have a right angle in there somewhere. And one thousand nine hundred and eighty have an obtuse one. Three groups, no overlaps, nothing left over. Set that beside the failed version, which collected all two thousand seven hundred and told you nothing.

6:17And here is that whole second sorting again, this time as a single movement. Two sides of five centimetres, hinged at the top, and the angle between them opening. At forty degrees, the other two corners are seventy each. Everything under ninety. Acute-angled. Keep opening. Eighty at the top, fifty and fifty below. Still acute, but only just. Ninety at the top. Forty-five and forty-five. Right-angled, for exactly one setting of the hinge.

6:52One more degree and it is obtuse, and it stays obtuse the rest of the way. The two lower corners are always the smaller pair, so the hinge on its own decides which of the three names this triangle gets. Now put the two sortings side by side, and see what they do to each other. Three side classes down one edge. Three angle classes along the other. Nine boxes, and a triangle belongs in exactly one of them.

7:25Scalene and right-angled: three, four and five, giving angles of thirty-seven, fifty-three and ninety. Isosceles and right-angled: two sides of five and a third a little over seven. Forty-five, forty-five, ninety. Isosceles and obtuse-angled: five, five and nine, which opens the top corner to a hundred and twenty-eight. Scalene and acute, isosceles and acute, scalene and obtuse. All built, all measured. Seven of the nine boxes, each with a triangle in it you could actually draw.

8:07Which leaves two boxes empty, and both of them are equilateral. Can an equilateral triangle have a right angle? Can it have an obtuse one? Do not reason about it, because reasoning about it needs something we have not got. Build one instead, and measure it. Three sides of seven. Measure the corners. Sixty, sixty, sixty. Three sides of two and a quarter. Sixty, sixty and sixty again. I built four thousand of them, at every size from a quarter of a centimetre upwards. Sixty every time.

8:47Never ninety, never past it. So those two boxes stay empty, and an equilateral triangle is always acute-angled. Though be careful what that is. It is four thousand measurements, not an argument. Why equal sides should force equal angles is a job for another day. One last question, and it is a counting one. A triangle with a right angle at B, and the side facing it, A C, five centimetres long.

9:21How many triangles are there that fit that description? The right angle takes ninety. The other two corners share the ninety that is left, and they can share it any way at all. Eighty-nine and one. Sixty and thirty. Forty-five and forty-five. Forty-five point one and forty-four point nine. There is no smallest step between two angles, so there is no last one on the list. The answer is not a number. There are endlessly many.

9:54And here is something you might notice while drawing them. Every one of those right-angled corners lands exactly two and a half centimetres from the middle of the five. Two thousand of them, all the same distance. Why, is another day's work too.