Decomposing an everyday object into the basic solids

0:00Here is a spinning top. Five centimetres tall, three and a half across the widest part, and there is no rule anywhere for the surface of a spinning top. There never will be. Nobody is going to hand you a formula for a top, or for a capsule, or for a rocket, because those are things people make, not shapes anyone chose to name. And yet this one is completely measurable, with nothing you do not already have.

0:31Because it is a cone, with a dome sitting on it. Two shapes you can already measure, put together. Seeing that is the whole of the mathematics here. Not a step before the mathematics - the mathematics. Start with what you were handed. A box, a cone, a cylinder, a ball. Four shapes, and the striking thing about them is how little it takes to pin one down. A box needs three numbers. A cone needs two. A cylinder needs two. A ball needs one.

1:06Now look at the end of a road tanker, and try to call it one of those four. It is not a cylinder: a cylinder holds the same width all the way along, and this narrows. It is not a cone: a cone narrows in a straight line, and this one curves. And it is not a ball. A ball stands as tall as it is wide. This stands exactly half as tall as it is wide, because it is half a ball.

1:36So the stock has to grow by one on the first real object you look at. The hemisphere - and you will use it more than anything else. Take the tank itself. A long cylinder, closed off with one of those domes at each end. Draw the three pieces apart and it is obvious. Push them together and it is still obvious. But the real thing has no line where the dome meets the cylinder. It is welded, and ground, and painted, and the seam is gone.

2:10That is why recognising the pieces takes an act of imagination rather than an act of reading. The finished object hides exactly the information you need from it. A test tube is the same move on something you have held: a cylinder, one rounded end. So the method is not a new technique. It is a way of not needing one. You are handed a problem you have never seen. You break it into problems you have already done. You do those. You put the answers back together.

2:47The only new thing being asked of you is the first step, and the first step is looking. Which raises a fair question. If I say the top is a cone and a dome, is that a fact about the top, or a story I told about it? It has to mean two things, and both of them can be checked. Nothing counted twice: the cone and the dome must not run into each other. And nothing missed: between them they must account for every point of the top.

3:21Here is the thing worth doing rather than assuming. Do not put the dome where you think it goes. Try every height, and keep the ones where the two pieces neither overlap nor leave a gap. Slide it down and the dome bites into the cone: they overlap. Slide it up and it floats clear: there is a gap. In between, exactly one position works. And on this top that one position sits three and a quarter centimetres up. The height of the cone was never something you were told. It is the only place the pieces will meet.

4:01But joining two things costs you something, and it is worth seeing the cost on an object simple enough to count. Two cubes, side by side. Each one is sixty four cubic centimetres, so each edge is four. Count the outside of one cube in square centimetres and you get ninety six. Two of them, apart, show a hundred and ninety two. Push them together and count again. A hundred and sixty.

4:30Thirty two square centimetres have gone. And thirty two is exactly twice sixteen - the face they were pushed together on, counted once for each cube. That is the rule for every join in this whole business. The two faces that touch stop being outside. They were, and now they are not. Now the difficulty that actually costs marks, and it is not a difficulty about shapes at all. Every number you are given describes the finished object. Total height. The width across the fattest part.

5:08Every rule you own wants a dimension of one piece. Those are almost never the same number, and the join is what converts one into the other. It makes the part heights add up to the whole - and where the object is meant to be smooth, it forces the two touching radii to be one radius. So before any formula appears, there is a translation step, and it is worth writing down rather than doing in your head.

5:39The smallest version of that translation catches more people than anything else here. Measure the top across its widest part and you get three and a half. That is the whole way across, so the distance from the axis - the radius every formula wants - is one point seven five. Now a bird bath: a cylinder with a hollow shaped into the top of it. Its given number is also thirty, and thirty is the radius here. Measured across, that bath is sixty.

6:14Same kind of number, opposite meaning, and the only way to know is to look at what the drawing marks. Halve the one you should not have halved and every area after it is wrong by a factor of four. So take the top all the way through the translation, and count what has to be manufactured. You are given two numbers. Five centimetres tall. Three and a half across.

6:42First: the radius. Three and a half across is one point seven five from the axis, and because the join is smooth that one number is the cone's radius and the dome's radius at once. Second: the dome is half a ball, so it stands as tall as its own radius. It takes up one point seven five of the five. Which leaves three and a quarter for the cone. That is where three point two five came from - the same number the search found on its own.

7:16Third, and this is the one people forget entirely: the slant. A cone's curved surface is built from its slant - the distance from the point to the rim, straight down the side. And the slant is essentially never one of the numbers you are given. It has to be built, out of the radius and the upright height, by Pythagoras. For this top: one point seven five across, three point two five up. Squared and added, that is thirteen point six two five.

7:51Its root is between three point six nine and three point seven zero, which is why the answer gets written as three point seven. But three point seven squared is thirteen point six nine, and that is not the number we started from. Three point seven is a rounding, and anything you build on it is a rounding too. Sometimes you get lucky. A cone of radius two and a half, six high, gives a slant of exactly six and a half; one of radius three and a half, twelve high, gives exactly twelve and a half. That is the arithmetic being kind, not the method being different.

8:37Now the object that exists to stop you over-learning the last lesson. A wooden rocket: a cone standing on a cylinder. Twenty six centimetres overall, the cone six of them. The cone's base is five across. The cylinder is three across. Those do not match, and they are not supposed to. The cone overhangs the cylinder all the way round. Measure the width just below the join and just above it, and it steps by exactly one centimetre. On the top, the same measurement gives nothing at all: no step.

9:16That is the real difference between the two objects, and it is a difference you can see. Smooth means the radii were forced to agree. Stepped means they were not. The heights still add, though: twenty six less six is twenty, so the cylinder is twenty tall. Adding up is what the join always gives you. Sharing a radius is what it gives you only sometimes. One more distinction, and there is a test for it that needs no arithmetic.

9:49Some of these objects have a piece stuck on. Some have a piece taken out. A dome fixed to the top of a cube is one; a bird bath with a hollow shaped into it is the other. Run a line straight up the middle of the object and ask, at every height, whether you are inside the solid. Stick a piece on and you are inside all the way up. Every height, without exception.

10:17Scoop a piece out of the top and the line leaves the solid before it gets there. On that bird bath, a hundred and forty five tall with a hollow of radius thirty, the middle gives out at a hundred and fifteen and the last thirty centimetres are a ring. Drill a cone right through a cylinder and the middle is never inside at all. Three objects, three answers, one test - and it is the same decomposition either way, run with a different sign.

10:50Before any of that, one thing has to be settled, and it will not settle itself. That bath was described as one point four five metres tall with a hollow of radius thirty centimetres. Two units in one sentence. Leave them mixed and the object is impossible. A dome of radius thirty does not fit inside a cylinder one point four five tall. It is not close. Convert first: one point four five metres is a hundred and forty five centimetres, and now the dome fits with room to spare.

11:29Then convert back at the end, if the question asked for metres. Mixed units do not come out in the wash. Last thing, and it is about a join that is not circle to circle. A dome on a flat square face is not matched to anything. It just has to fit. On a face seven centimetres square the biggest dome you can set down has a radius of three and a half, because the circle has to stay inside the square and the square gives you only half its edge in every direction.

12:06On a five centimetre face, a dome four point two across leaves four millimetres of clearance all round. It fits, and you should be able to say why it fits. That block, by the way, is not a shape you can turn about a line at all - and its height still adds up, five plus two point one, and the widest thing about it is still the cube. Which is the honest scope of all this. Two basic solids at a time, joined or hollowed. It is not a general theory of shapes. It is a habit of looking at a made thing and seeing what it was made out of.