Triangles and regular polygons: when equal sides let you multiply

0:00A triangle. Five centimetres up one side, four up the other, seven along the bottom. What is its perimeter? Same as always — go round the edge and add. Five, plus four, plus seven. Sixteen centimetres. And notice what did not happen. No shortcut. No formula. No doubling, no multiplying. Because nothing repeated. Three sides, three different numbers, so the sum stays a sum. That is worth starting with, because it is the normal case. Most triangles look like this one.

0:37The shortcuts we are about to build are the exception — a mercy that certain shapes happen to grant you. So the triangle rule is simply this: perimeter equals the sum of the three sides. That is all it says. Which sounds like it is not doing any work. But run it backwards, and it is. A triangle has a perimeter of fifty-five centimetres. Two of its sides are twenty and fourteen. Find the third.

1:09Twenty plus fourteen is thirty-four. Fifty-five minus thirty-four is twenty-one centimetres. You just found a length you could not see, from a total and two parts. That is what a rule is for. Forwards it saves you very little. Backwards it tells you something you did not know. Now a triangle where something does repeat. Triangle A B C, with all three sides the same length. The book calls it equilateral. Equal-lateral — equal sides.

1:44Start exactly where we always start. Perimeter is A B plus B C plus C A. But B C is the same length as A B. And so is C A. So replace them. A B, plus A B, plus A B. The same number, three times over. And three of the same thing added is that thing times three. So the perimeter of an equilateral triangle is three times one side. If the side is five, the perimeter is fifteen.

2:16Notice we did not learn a new fact. We noticed a repetition and gathered it up. And you have seen that exact move before, one section ago. Here is the square, done the same way. Perimeter is side, plus side, plus side, plus side. All four are equal, so it is one number four times over — four times the side. Put the two derivations beside each other and they are the same sentence with a different count.

2:48Three equal sides became times three. Four equal sides became times four. So the book asks a very good question here. What does an equilateral triangle have in common with a square? Hold that for a moment before answering, because the answer is the whole topic. The book's answer is a name. Both of them are regular polygons. And regular here is a technical word with two halves, so let us say both halves out loud.

3:22A regular polygon has every side the same length. And every angle the same size. Both conditions, not one. An equilateral triangle has three equal sides and three equal angles. A regular pentagon has five equal sides and five equal angles. And regular does not mean ordinary, or usual, which is what that word means everywhere else in your life. In this chapter it is a test a shape either passes or fails, and it has two parts.

3:58Keep both parts in mind — because in a few minutes we are going to find that the shortcut only needs one of them. You have met this family before, in the very first chapter of this book, in the shape sequence. A triangle, a square, a five-sided pentagon, a six-sided hexagon — each with all its sides equal and all its angles equal. Line them up with the same side length, five centimetres each, and walk round every one.

4:32The triangle: three fives, fifteen. The square: four fives, twenty. The pentagon: five fives, twenty-five. The hexagon: six fives, thirty. Now look down that column of answers. Fifteen, twenty, twenty-five, thirty. Each one is five more than the one above it. Of course it is — each shape has one more side, and every side is five. And look at how each answer was built. Three fives. Four fives. Five fives. Six fives.

5:09The side count, times the side length. Every single time. So a regular seven-sided figure with sides of five centimetres would be thirty-five, and you did not need to see the shape to know that. There is a general rule forming in front of you. But be careful here, and be honest about what has just happened. The book works this out for three sides. It works it out for four. Then it stops.

5:39It does not print the general rule anywhere in this chapter. It asks you to find it, and it asks your teacher to help you find it. That is not an oversight. Two cases that agree are a pattern, and a pattern is an invitation, not a proof. You are meant to test it. Draw a regular pentagon, measure one side, multiply by five — then lay a string round the edge and check.

6:07The rule will survive that. Do it anyway. Being handed the answer would have cost you this. Here is a test that comes at the same rule from the other end. One piece of string, thirty-six centimetres long. Bend it into a square. Four equal sides sharing thirty-six. Thirty-six divided by four — nine centimetres each. Now unbend it and make a triangle with three equal sides. Thirty-six over three — twelve centimetres each.

6:41Now a hexagon: six equal sides. Thirty-six over six — six centimetres each. Same string, same perimeter every time, and yet the side gets shorter as the sides get more numerous. The book picked three, four and six because they divide thirty-six neatly. A regular pentagon works just as well — thirty-six over five is seven point two centimetres. A real shape, just not a whole number. Now the sharpest question in this topic, and the book leaves it to you.

7:20The shortcut needs equal sides. Does it also need equal angles? Here is a four-sided figure with four sides of two centimetres each — but pushed over, so its corners are not square. It is not a regular polygon. It fails the angle half of the test outright. But go round it. Two, plus two, plus two, plus two. Eight centimetres. Four times the side, exactly as before. Push the corners as far over as you like. As long as those four sides stay two centimetres, the perimeter stays eight.

7:59The angles never entered the calculation. A perimeter only ever adds up lengths. And it fails the other way round too, which settles the matter. A rectangle, six by two. Every angle is a right angle, so it passes the equal-angles half perfectly. Try the shortcut anyway. Four times six is twenty-four. The true perimeter is sixteen. Wrong, and badly wrong, because the sides were never all equal. So put the two together. Equal sides but not equal angles: the shortcut works. Equal angles but not equal sides: the shortcut fails.

8:44Only one of the two conditions is doing any work at all. Being regular is enough to guarantee the shortcut. It is more than the shortcut actually needs. So here is the rule, and it is yours to write, because the chapter does not. For a polygon whose sides are all equal, the perimeter is the number of sides, times the length of one side. And say the condition out loud every time you write it. All sides equal. Not regular — equal-sided. That is the half that matters.

9:22Then the task the chapter ends on. Find regular-shaped things around you — a floor tile, a nut on a bolt, a stop sign, a paving slab. Measure one side, count the sides, multiply. Then run a tape round the edge and see whether you were right. That is the whole of perimeter: go round the edge, add, and take a shortcut only when the shape has earned you one.

9:49Next time we leave the boundary behind altogether, and ask about the inside.