Which faces vanish at the join, and why you cannot simply add

0:00Here are two solids, and here is a tin of paint. Paint the cone. Every part of it: the sloping side, and the flat circle it stands on. Now paint the dome. The curved skin, and the flat circle underneath it. Add the two amounts up, and you have a number. Now put the dome on the cone, and paint that. The number is smaller. It has to be. Because the circle on the bottom of the dome and the circle on the top of the cone are pressed together, and you cannot get a brush in there.

0:35Two circles of paint, bought and never used. That is the whole of this lesson, and everything else is working out exactly which paint gets wasted, because it is not always two circles. So it is worth being precise about what surface area is. It is not a property of the material. It is a property of the boundary. Surface area measures the set of points you could put paint on: the points that are on the solid and have nothing but air just beyond them.

1:10Everything inside is not surface. It never was. And that gives you a test you can apply to any point of any object, one point at a time. Stand at a point. Is it on the solid? Step a hair outwards. Are you still on the solid? If you are on and then off, that point is outside, and it counts. If you are on and then still on, you are in the middle of the material, and it does not.

1:38Every question in this topic is that one test, asked at the right places. Watch what the test says as two pieces come together. Here is a cylinder with a flat circular end, and a dome with a flat circular face. Apart, both of those flat faces are outside. Stand on either one, step outwards, and you are in the air. Now slide them together. The instant they touch, every point of those two circles fails the test. Step outwards from the cylinder's end and you are inside the dome. Step outwards from the dome's face and you are inside the cylinder.

2:22The two faces have not gone anywhere. They are still there, made of the same material. But they have stopped being boundary, and boundary is the only thing surface area was ever measuring. On the finished object there is no seam to see at all, and that is the difficulty: the picture hides exactly the information you need. Start with the easiest case, where the two faces are the same circle laid exactly on top of each other.

2:55A capsule: a straight tube with a rounded end at each end. Fourteen millimetres long, five across. Five across means a radius of two and a half. The two rounded ends take up two and a half each, so the straight part is nine. There are four flat circles inside that object, and every one of them is pressed against another one, so not one survives. What is left is the curved wall and the two curved caps.

3:26That comes to two hundred and twenty square millimetres. And here is something worth noticing. Two hundred and twenty is exactly what you would get from the curved wall of a plain tube fourteen millimetres long and five across. The two domes together give back precisely what rounding the ends took away. Now the object that shows you the cost of getting this wrong. A spinning top: a cone standing on its point, with a dome on top of it. One point seven five from the axis, three and a quarter of cone, and a slant of three point seven.

4:07Its surface is thirty nine point six square centimetres. Add the two pieces' surfaces instead, the way the first instinct says, and you get fifty eight point eight five. The gap is nineteen point two five, and that number is not mysterious. The circle where they touch measures nine point six two five. You counted it once as the cone's base and once as the dome's flat face, and neither one is on the outside of anything.

4:40Nineteen point two five is exactly twice nine point six two five. Not roughly. Exactly, and that is the shape of the correction. So write the rule down properly. The outside of a joined solid is the two outsides added together, less twice the region where the pieces are in contact. Twice, because each piece was carrying its own copy of that region and both copies have to go. And read that again, because there is a trap in it.

5:15It says twice the CONTACT. It does not say two circles. The contact is whatever the two faces actually have in common, and that depends entirely on how they meet. If they match, the contact is the whole circle and the whole of both faces goes. If a small circle sits on a big flat face, only that small circle goes and the rest of the big face is still out in the air.

5:43And if two circles of different size are pressed together, the contact is the smaller one, and a ring of the bigger one survives. Three cases. One rule. Take the second case, and predict the answer before you compute it. A cube five centimetres on a side, with a dome sitting on its top face. The dome is four point two across, so its radius is two point one. The bare cube has six faces of twenty five, which is a hundred and fifty.

6:19Now put the dome on. Most people expect the total to go down: you have covered something up. It goes up. The circle you covered measures thirteen point eight six, so the top face keeps twenty five less thirteen point eight six. But the dome's curved skin measures twenty seven point seven two, which is exactly twice the circle it stands on. A hemisphere always is. Lose one circle, gain two. The block comes to a hundred and sixty three point eight six, which is thirteen point eight six more than the cube it was made from.

7:01Now the third case, and the one that punishes a memorised rule. A wooden rocket: a cone standing on a cylinder. The cone's base is five across; the cylinder is only three across. Look at it from directly above and you see two circles, one inside the other. The inner one, radius one and a half, is where the two pieces are actually in contact. That is the part that stops being outside.

7:31The ring between them is not in contact with anything. It is the underside of the cone's overhang, and it is out in the open air, and it needs painting like everything else. That ring measures pi times two point five squared less one point five squared, which is four pi, twelve point five six. Subtract two circles here and you have thrown away a piece of the object that is plainly visible from underneath.

8:04The rocket is usually asked in two colours, which makes the audit explicit. The cone gets one colour. Its curved side, using a slant of six point five, and the ring underneath it. Sixteen point two five plus four, times pi: sixty three point five eight five. The cylinder gets the other. Its curved wall, and the circle it stands on at the very bottom, which nothing covers. Its own top face gets nothing, because the cone is sitting on all of it.

8:39Sixty plus two point two five, times pi: one hundred and ninety five point four six five. Two faces of the same solid, handled two different ways in one question. The top is hidden; the bottom is not. And the two amounts add to the whole outside of the rocket, which is how you know the audit missed nothing. Everything so far has been about sticking pieces on. Now take one away.

9:10A cylinder two point four tall and one point four across, with a cone of exactly the same height and width drilled right out of it. Which faces are outside now? The curved wall of the cylinder, still there. The circle it stands on, still there. Its top face is gone completely, because the cone's mouth is the same width as the cylinder. And in place of it there is a new surface that did not exist before: the inside of the cone, curved, with a slant of two point five.

9:47Seventeen point six square centimetres, which is eighteen to the nearest. Drilling a hole out of something removed material and made more surface, not less. Boundary was created. The same on a cylinder with a bowl hollowed into each end: three hundred and seventy four. Which sets up the fact that makes the whole topic click. Take a cube seven centimetres on a side, and the largest dome its face will take, radius three and a half.

10:20Stick that dome on. You lose the circle it covers and you gain twice that circle in curved skin, so the surface goes up by one circle: two hundred and ninety four plus thirty eight point five, three hundred and thirty two point five. Now start again, and scoop the identical bowl into the face instead. You lose the same circle from the flat face, and you gain the same curved skin, now facing inwards.

10:52Three hundred and thirty two point five. The same number. Adding a dome and hollowing an identical bowl change the surface by exactly the same amount, because the bookkeeping is identical and only the sign of the material changed. One more thing decides the answer, and it is not geometry. A bird bath: a cylinder a hundred and forty five centimetres tall, with a bowl of radius thirty hollowed into the top.

11:26Its whole boundary includes the flat underside. The published answer does not: three point three square metres, with the underside left out, because the bath stands on legs. A tent behaves the same way. Cylindrical wall two point one metres high, four metres across, conical top of slant two point eight. Forty four square metres of canvas, and at five hundred a square metre that is twenty two thousand. The floor is not canvas, and the question says so.

12:02And a vessel has an inside as well as an outside. Asked for the inner surface of one thirteen deep and fourteen across, you want five hundred and seventy two, leaving out the mouth. Deciding which faces the question wants is part of the problem, not a detail before it. A word about the numbers themselves, because two of them are not what they look like. The top's slant is not three point seven. It is the square root of thirteen point six two five, which is three point six nine one two and does not stop.

12:42Three point seven squared is thirteen point six nine, not the number we started from. Everything built on it is a rounding. The value used for pi changes the last digits too. Twenty two over seven and three point one four are not the same number. So a good answer says which one it used. But notice what does not depend on any of that. The overcount on the top was exactly twice the contact circle. The block gained exactly one circle. The dome and the bowl came to exactly the same. Those are the facts of the topic, and no rounding touches them.

13:27Which leaves a procedure that works on anything, including something you have never seen. List every face and every curved skin the pieces have. Against each one, ask the paint question: is there air just beyond it? Mark it seen or hidden. Then add up the seen ones. That is it. There is no formula for a rocket or a bird bath or a capsule, and there is never going to be one.

13:55Two cubes make the point in miniature. Sixty four cubic centimetres each, so each edge is four, so each shows ninety six square centimetres. Apart, a hundred and ninety two. Pushed together, a hundred and sixty. Thirty two gone, which is twice the sixteen of the face they touch on. Not a formula. A count of what is still on the outside.