Why the angles of any quadrilateral add to 360°

0:00Here is a question that sounds like it should have an easy answer. Can you draw a four-sided figure with three square corners, and a fourth corner that is not square? Nothing about that sounds impossible. Three corners at ninety, and the last one at, say, a hundred and twenty. Most people say yes, and then reach for a pencil to show you. Try it. Draw a line, turn ninety, draw another, turn ninety, draw a third, turn ninety.

0:31Now the fourth side has to get you home, and you will find it already has. The figure closes itself, and it closes at ninety, and you were never asked. So put the fourth corner where you wanted it. A hundred and twenty. Turn a hundred and twenty instead of ninety and the last side sets off in the wrong direction. It misses the corner you started from, and no length of it will ever reach.

1:00You can slide the sides around, lengthen them, shorten them. It never closes. And that is a strange thing to be stuck on, because nothing has gone wrong yet. Three ninetys is not a lot to ask of a four-sided figure. The reason it refuses is not about drawing. It is about a total, and the total is not free to be anything. A four-sided figure looks like it needs a rule of its own.

1:30It does not. It needs one line. Join two corners that are not next to each other. That line is a diagonal. The moment it is there, the figure is not a four-sided figure any more. It is two triangles. And you already know something about a triangle that you had to be told once and never again. Its three angles total a straight angle. One hundred and eighty. That is the only fact this whole thing uses. There is not a second one coming.

2:05Label the corners so we can keep track. Call them S, O, M and E, going round. The diagonal runs from S to M, and it cuts the figure into a triangle holding E and a triangle holding O. Number the angles the diagonal creates: one, two and three in the lower triangle, four, five and six in the upper. One plus two plus three is a hundred and eighty.

2:34Four plus five plus six is a hundred and eighty. Add them and the six angles total three hundred and sixty. Which looks like the answer, and is not, because six angles are not four corners. This is the step almost everybody walks straight past, and it is the entire argument. Angle two sits alone at E, so it is that corner already. Angle five sits alone at O, same.

3:04But at S there are two of them. Angle one is below the diagonal and angle four is above it. Neither of those is the corner at S. The corner at S is both of them together, with the diagonal running through the middle. Slide the diagonal out and watch one and four close up into a single angle. The same thing happens at M, where three and six rejoin.

3:31That regrouping is why six numbers are allowed to become four, and without it the sum is just six angles that happen to add up. So the four corners of the figure total three hundred and sixty. Not the corners of that figure. The corners of any of them. Here is a family of four-sided figures built by walking one corner over a grid, with nothing selecting for a nice shape.

4:01A hundred and thirty-seven of them come out, wide ones, thin ones, and twenty-one that cave inwards. Total the four corners of every single one. Across all one hundred and thirty-seven, the total takes exactly one value, and the value is three hundred and sixty. The two hundred and seventy-four triangles that diagonal cuts out all total a straight angle, every one, which is the same fact wearing a different hat.

4:34Now the drawing that refused to close makes sense. Three square corners spend ninety, ninety and ninety out of a budget of three hundred and sixty. What is left for the fourth corner is ninety, and there is nothing to negotiate with. Not ninety by tradition. Ninety because the other three took everything else. Check it on nine rectangles of nine different widths and the fourth corner is ninety in all of them.

5:04So three square corners force the fourth, and a four-sided figure with exactly three of them cannot exist. Which also answers a question people ask straight afterwards: three square corners do make it a rectangle. There is a wrong turn here that is worth taking on purpose. A four-sided figure has two diagonals. If one gave two triangles, why not draw both and get four? Draw both. Four triangles, twelve angles, and twelve angles total seven hundred and twenty.

5:42Which would make the figure's corners total seven hundred and twenty, and that is plainly wrong. So find the extra. The diagonals cross at a point in the middle, and four angles sit round that point. Those four are not corners of the figure at all, and they total three hundred and sixty, on all one hundred and sixteen shapes where both diagonals cross inside. Take them off the seven hundred and twenty and you are back where you started. The second diagonal added nothing but bookkeeping.

6:20Now the case that breaks careless versions of this argument. Take a four-sided figure with a corner pushed inwards, so it dents rather than bulges. It is still four straight sides and four corners. It is still a quadrilateral. Draw a diagonal and cut it into two triangles, exactly as before. Except that one of its two diagonals does not stay inside the figure. It runs out through the dent and back in.

6:50That line cuts nothing into anything. The two-triangle argument cannot be run along it. Of the twenty-one caved-in shapes in the sweep, every one has exactly one usable diagonal, and the other hundred and sixteen have two. There is a second trap at the dented corner, and it is the one that makes students think the rule has failed. At that corner two angles meet the same pair of sides: a small one on the outside and a large one on the inside.

7:26The figure's corner is the large one. That is where the inside of the figure is. On this shape the four corners measure sixty-three point four three, thirty point nine six, two hundred and thirty-nine point zero four, and twenty-six point five seven. They total three hundred and sixty, and the rule has not moved. Lay a protractor down without thinking, read the small angle — a hundred and twenty point nine six — and the total comes to two hundred and forty-one point nine three.

8:03That is short by a hundred and eighteen point zero seven, which is the difference between the two angles at that one corner. It is fair to ask whether any of this depends on how we wrote the figure down. So run the whole sweep again with every shape listed from the next corner round. Same shapes, same order, different starting letter. Now the diagonal that stays inside is the other one of the two, on all twenty-one that cave in.

8:36Every shape still regroups, and every total is still three hundred and sixty. Write a figure down backwards and its four angles come back in reverse order without one of them changing. Which diagonal you may draw is a fact about the shape. Which corner you called the first one is a fact about you. And measuring it the naive way is not a small error: across the sweep the wrong measurement reports thirteen different totals, the lowest of them two hundred and twelve point five.

9:13So nothing was discovered here. There is no rule for four-sided figures. There is a rule for triangles, and a line that turns one problem into two of those. Three hundred and sixty is not a number to remember. It is one hundred and eighty, used twice. A student who has seen the diagonal never has to memorise it, and can rebuild it on a bus. And the same move keeps working. Five sides, three triangles, five hundred and forty. Six sides, four triangles.

9:48Each new side buys one more triangle and one more straight angle, and it is the same one line doing the work every time. The best thing a piece of mathematics can be is an old thing you already had, pointed somewhere new.