Regular hexagons, and why the angles round a point must total 360°

0:00Here is a hexagon, and every one of its six sides is exactly the same length. So is it a regular hexagon? Look at the corners. Two of them are dead square. The rest are not, and they are not the same as each other either. So, no. Equal sides is only half of it. Regular means equal sides and equal angles, both, and forgetting the second half is the easiest mistake here.

0:30Here is one that really is regular, and you can see the difference without measuring a thing. Regular shapes, listed by how many sides they have. Three is the equilateral triangle, and that one is already in hand. Three equal sides, and each corner a third of a straight angle, which is sixty. Four is the square. Four right angles, and the right angle you already know how to build.

1:00Five and six are where it gets interesting. Try drawing each of those by hand, with all the sides equal and all the angles equal, and see how you get on. One of them is going to come out. The other is going to be left for later, and not because it is harder to draw neatly. Take the regular hexagon as given for a moment, and open it up.

1:26Mark the centre — not just any point in the middle, but the one that every corner is the same distance from. Join that centre to all six corners. Six triangles, and they fill the hexagon exactly, because no two of them overlap and none of it is left out. Now look hard at what those triangles actually are. Every corner sits the same distance from the centre, and that is not luck. It is what the word regular is doing.

1:58So all six spokes are one length, and each triangle has two equal sides before you have looked at anything else. The third side of each triangle is a side of the hexagon itself. And on this figure that third side comes out equal to the two spokes as well. Which is not obvious, and for the moment it is an observation rather than an argument. But take it. Three equal sides, so every one of the six triangles is equilateral.

2:32Which means all three angles inside it are the same angle as each other. Three equal angles sharing a straight angle between them, so sixty each. That is the whole hexagon, taken apart — and now the argument, which runs the other way round. Now turn the question round, because this is the move that does all the work. Forget that you ever had a hexagon. Start instead with six equilateral triangles, all the same size.

3:06Bring them corner to corner and lay them round a single point. Do they close up? If they do, and the sixth one comes back exactly onto the first with nothing overlapping and nothing left uncovered, then look at what you are holding. Six equal sides on the outside, and at every outer corner two triangles meeting, sixty and sixty. What you are holding is six equal sides, and at every outer corner two sixties, which is a hundred and twenty.

3:41Equal sides and equal angles. A regular hexagon, and you never drew one. You assumed six equilateral triangles, and the hexagon arrived on its own. So the whole question has moved somewhere else entirely. It is not about hexagons any more. It is about whether six sixty-degree corners can be crowded round one single point. And a question about angles at a point is a far easier thing to settle than a question about shapes.

4:16Angles at a point, then. Start along one direction, sweep all the way round, and arrive back where you began. That is a complete turn. A complete turn is three hundred and sixty degrees. Not by discovery, but by definition. A degree is one three hundred and sixtieth of the whole way round, and that is where the number comes from. Which makes a full turn the total that every fan of angles at a point has to come to, whatever those angles happen to be.

4:51So the question stops being a matter of judgement and becomes arithmetic, and arithmetic is something you can be certain about. Here is a point with wedges fanned out round it, and one gap left over. Each wedge is tagged with the angle it takes up. Forty. Sixty. Fifty. Thirty. Forty. And ninety. And here, off to one side on its own, is a piece tagged seventy. The question is whether it will go into the gap.

5:24Look at it for a moment. It looks about right, doesn't it — near enough that you would probably try it. Do not answer by looking. That is the whole reason this puzzle is sitting here. Add up the six that are already down. Forty and sixty is a hundred. Fifty more is a hundred and fifty. Thirty makes a hundred and eighty. Forty makes two hundred and twenty. Ninety makes three hundred and ten.

5:55The whole way round is three hundred and sixty. So the gap is fifty. And the piece is seventy. It is twenty degrees too big. It does not go in, and no amount of turning it about will help. The only piece that fills that gap is fifty. Smaller leaves a gap of its own, and bigger will not go. Which gives you the rule, and it is worth saying in a single line.

6:23Angles brought together at a point fit, with nothing overlapping and nothing uncovered, exactly when they add up to a full turn. That is the whole of it. Notice what it does not mention. It says nothing about what the pieces are, or what shape they came off. It is a statement about angles at a point, which is exactly why it can settle the hexagon without ever saying the word.

6:53So. Six sixties. Six times sixty is three hundred and sixty. They fit, exactly, with nothing left over. The six triangles close up, the corners come out equal, and the regular hexagon is real. And here is something else that falls out of it, free. Take three of those sixties in a row at the centre. Three sixties is a hundred and eighty, and a hundred and eighty is a straight angle.

7:23So each of those long lines through the middle is genuinely straight, corner to opposite corner. Not nearly straight, and not straight because it looks it. Now go back to the pentagon. Same idea. Five triangles from the centre, meeting at a point. Five wedges share the full turn between them, so each one is three hundred and sixty divided by five. That is seventy-two. And seventy-two is not sixty, so that triangle is not equilateral. Its base comes out shorter than its two spokes.

8:03The entire argument rested on the wedge being sixty, and a full turn divided by six is the only way to get one. So the pentagon is not out of reach for ever. It is out of reach by this route, and it waits. Building a hexagon is now almost nothing. Set the compass to the side you want, and draw a circle of that same opening. Step the compass round the circle, marking as you go.

8:34The sixth step lands exactly back on the first, and it lands there for the same reason the triangles closed up. Join the marks, and there it is. One last thing, and it costs nothing either. Stand an equilateral triangle outward on each of the six sides. Every one of those points has three equal sides, because the side of the hexagon and the two you drew are all one length, and what you are looking at is a six-pointed star.

9:07The hexagon was never the difficult part. The point in the middle was.