Why every equilateral triangle has three 60° angles

0:00Equilateral. Three sides, all the same length. That is the whole of what the word says. It is a claim about lengths. Now notice what it does not say. It says nothing at all about the angles. So here is a triangle with three equal sides, and three angles we know nothing about. You have probably been told they are all sixty degrees. And they are. Every equilateral triangle, at every size, every single one.

0:31But that is not part of the definition, and nobody here is going to measure it. It has to come out of an argument. So let us squeeze it out. We are going to need one result, and we already have it. In any triangle, if two of the sides are the same length, the angles facing them are equal. That was proved. Not measured, proved, by cutting the triangle in half and matching the pieces.

1:01So it applies here. And it applies as often as we care to use it. But look at what it needs. It needs a pair of equal sides. Point at a pair, and it hands you back a pair of equal angles. Now, an equilateral triangle has three sides, and all three are equal. Which means there is more than one pair to point at. That is the whole idea.

1:28Before we use it, we have to be exact about one word. Facing. Here is the triangle, lettered A, B and C. The angle at A sits between the two sides A B and A C. So it does not face either of those. It faces the third one, B C. Same everywhere else. The angle at B faces A C. The angle at C faces A B. Every angle faces the side made from the other two letters.

1:58This is worth slowing right down for, because getting it backwards loses everything after it. The angle at A does not face A B. It faces B C. Right. First pair. Mark the sides A B and A C as equal. They are, of course. All three are. But those two are the ones we are attending to. Now apply the result. Two equal sides, so the angles facing them are equal.

2:28Which angles are those? The one facing A B, and the one facing A C. The angle facing A B is the angle at C. The angle facing A C is the angle at B. So the angle at B is equal to the angle at C. Two of the three, tied together. And nothing was measured to do it. One more pair, and we will have all of them.

2:54Now watch the figure closely, because the only thing about to change is the marks. The triangle stays exactly where it is. Same size, same position, same everything. Take the tick marks off A C, and put them onto B C instead. So the pair we are attending to now is A B and B C. Nothing about the triangle changed. Only which two sides we are pointing at. And that is allowed, because all three of them really are equal.

3:27This is the move that does the work, and it is very easy to walk past. One result, used a second time, on a different pair. So run it again. A B equals B C, so the angles facing them are equal. And take care here, because this is exactly where it goes wrong. The angle facing A B is the angle at C. That one has not moved. And the angle facing B C is the angle at A.

3:58Not the angle at B. B sits between the two marked sides, so it faces neither of them. So the pair we get this time is the angle at A and the angle at C. Angle A equals angle C. Different pair of sides. Different pair of angles. Same result doing the work. Now put the two of them together. From the first pass, the angle at B is equal to the angle at C.

4:28From the second, the angle at A is equal to the angle at C. Both of them are tied to that same angle at C. So if A matches C, and B matches C, then A has to match B as well. All three angles are equal to one another. And look at what that took. One result, used twice, on two different pairs. There is still no number anywhere. Just three angles that are obliged to match.

4:57Now the number. And this is the part that feels like a trap closing. The three angles of any triangle at all come to a hundred and eighty degrees. And we have just shown that these three are all the same size as each other. So three of that one size make a hundred and eighty. Three times something is a hundred and eighty. There is exactly one number that does that. A hundred and eighty, divided by three.

5:27Sixty. Every angle of every equilateral triangle is sixty degrees, and no other value is possible. Now, you could have got to sixty another way. You could have drawn one and measured it. And you would have read sixty, near enough. So why go to all this trouble? Because near enough is what a protractor gives you, and near enough is not the same thing as exactly. Take one marked to the half degree. Anything from fifty nine and a half to sixty and a half reads as sixty.

6:04Now here is a triangle whose angles are fifty nine point six, sixty point two, and sixty point two. Those add to a hundred and eighty, and every one of them reads as sixty on that scale. But its sides are not all the same length. They differ by about half a percent. So the measurement was happy with a triangle that is not equilateral. The argument was not. So go back and look at what the whole thing actually stood on.

6:37One result. Equal sides give you equal angles facing them. That result used twice, on two different pairs of the very same three sides. Then one fact that holds for every triangle. The three angles come to a hundred and eighty. Then one division. That is the entire list. And there is no protractor anywhere in it. The sixty was not observed, and it was not defined into being. It was squeezed out.

7:09By the end, there was nothing else left for it to be. One more thing worth being clear about, because it is easy to over-read. Sixty degrees is what all equilateral triangles share. It is not what makes them the same. Here is one with sides of one unit. Here is one with sides of five. Both equilateral. Both carrying three sixty degree angles. And they are not the same triangle. Not even close.

7:41Angles hand you the shape. They never hand you the size. So equilateral triangles come in every size there is. What they have in common is that sixty, and nothing else at all. Triangles that match each other exactly are everywhere, once you start looking for them. A glass pyramid, four identical triangular faces leaning in to meet at a point. A stone one, far older and far larger, built out of the same idea.

8:14A dome broken into a mesh of triangles. A floor pattern of coloured ones laid out as a star. A bridge, carried on a row of steel triangles repeating the whole way along. In every one of those the triangles match each other, and that is what makes them work. But be careful what you claim. Matching each other is not the same as having three equal sides. The points of that star do. Those pyramid faces do not, and neither do the bridge's.