Why volumes do add even though surface areas do not

0:00Here are two solids about to be joined: a cylinder, and a dome that fits it exactly. Measure the outside of each one, add the two numbers, and you have too much. The circle on top of the cylinder and the circle under the dome are pressed together, and neither is on the outside any more. So here is the obvious worry. If you measure how much space each piece takes up and add those two numbers instead, will you have too much again?

0:31No. Volumes add. Exactly, with nothing lost and nothing counted twice. Two measurements of one join, behaving in opposite ways. That looks like two separate rules to remember. It is one, and this video is the single sentence that settles both. Before adding anything, be clear about what the two numbers measure, because they do not measure the same object. Volume is the region a solid occupies. Every point of space inside it.

1:06Surface area is the skin. Only the points on the boundary, with air just beyond them. One is about the inside. The other is about the edge. Fill the solid with water and read the jug: that is the volume. Wrap it in paper and measure the paper: that is the area. Now push two pieces together, and ask what changed. The edge changed. Two faces that had air beyond them now have material beyond them.

1:37The inside did not change at all. Not one point of space moved, appeared, or went away. So look hard at the thing that changed hands: the flat face where the pieces meet. Take a square face, 4 across, and measure it twice. Cut space into little cubes, and cut the face into the squares those cubes stand on. At one fineness the face is 64 squares. At a finer one, 4096. The area they come to is 16 every single time.

2:14The face has an area, and the area is 16. Now count how many little cubes the face holds. None. Not one, at any fineness at all, because a cube needs thickness to sit in and a face has none. Give the face a thickness and watch. Thickness 1, and it holds 16 of room. A half, and it holds 8. A quarter, 4. An eighth, 2. Halve the thickness, halve the room, while the area stays at 16 through all of it.

2:46Take the thickness down to nothing, and the room goes with it. A face has area, and it takes up no room whatsoever. That one sentence settles both rules. Here is how. Join two solids along such a face, and ask where a point of space can be. In the first piece, in the second, or in neither. Nothing is counted twice, because all the pieces share is the face, and the face holds no space to count.

3:17Nothing is lost, because no point of space vanished when they touched. So the room the joined solid occupies is the two rooms added. That is the entire argument. Now read the same sentence for the other measurement. The shared face was part of one piece's boundary and part of the other's, and it is part of neither now. Area goes missing exactly where volume does not, for exactly the same reason.

3:49Watch both readings on something small enough to count by hand. Here is a cube built from little cells, 4 along every edge, so 64 cells in it. Here is a second cube just like it. Also 64. Push them together so they share a whole face, and count the cells in the pair. 128. Nothing left over and nothing counted twice. Now count faces instead. Each cube alone shows 96 little faces to the air.

4:23The pair shows 160, and 96 plus 96 is 192. 32 faces have stopped being outside, and the cubes are pressed together on 16. Twice the contact, on any pair that meets along a face. One set of cells, two readings off it. The room adds. The skin does not, and what it loses is twice what they meet on. The adding rule needs one condition, and it is easy to miss because the objects you meet never break it.

5:00The pieces must not overlap. Drive the second cube two cells into the first instead of setting it alongside. Now 32 cells belong to both. Count the pair: 96 cells, not 128. The sum is 32 too big, and 32 is exactly how many cells got counted twice. So the honest statement is this: volumes add when the pieces meet on a boundary and share nothing else. Solids joined face to face satisfy that automatically, because a face holds no room.

5:38It is a rule with a reason, not a slogan, and the reason tells you the one case where it fails. Time to use it. Here is a shed with a curved roof. The base is 7 metres by 15, the walls are 8 metres high, and the roof is half a cylinder lying along the top. The walls make a box: 15 times 7 times 8 is 840 cubic metres.

6:08The roof is half a solid, and the rule was stated for whole ones. Does that matter? It does not. Half a cylinder is itself a solid occupying space, and the flat rectangle where it rests on the walls has no volume either. And half really is half. Cut a cylinder by the plane straight through its axis and count the cells on both sides. 2340 in all, and 1170 on each side, one the mirror image of the other.

6:43The roof spans the 7 metre width, so its radius is 3.5, and it runs the full 15 metres. The whole cylinder measures 183.75 times pi, so half is 91.875 pi, which with pi as 22 sevenths is 288.75. Add: 840 and 288.75 give 1128.75 cubic metres. Then comes a second question, and it is not the same question. How much air is in the shed? That is the room the shed encloses, less the room everything inside it takes up.

7:32There are 300 cubic metres of machinery on the floor. And 20 workers, each occupying about 0.08 of a cubic metre. 20 times 0.08 is 1.6 cubic metres. Small against a thousand, and tempting to drop. Do not. Deciding that something is negligible is a decision you state out loud, not one you make quietly. 1128.75, less 300, less 1.6, leaves 827.15 cubic metres of air. The enclosure and the free space are different numbers, and reading which one is wanted is part of the problem.

8:23Here is a toy: a cone standing on a hemisphere. The cone is 2 centimetres tall and the base is 4 across, so the radius is 2 and the two pieces share it. Now draw the smallest cylinder that just encloses the whole toy, touching it all round. How tall must that cylinder be? Not 2. The hemisphere hangs its own radius below the join and the cone rises its own height above it, so the cylinder is 2 plus 2, which is 4.

8:57The toy first. Hemisphere and cone add, because they meet on a flat circle that holds no room. That comes to 8 pi, and with pi as 3.14 it is 25.12 cubic centimetres. The cylinder: radius 2, height 4, which is 16 pi, or 50.24. So the space left over inside the cylinder is 50.24 less 25.12. 25.12. The leftover is exactly the size of the toy sitting in it.

9:36That looks like luck. It is not, and it has nothing to do with the number 2. Take any radius you like, and let the cone's height match that radius. A hemisphere fills exactly two thirds of the cylinder that just holds it. A cone fills exactly one third of the cylinder that just holds it. Because the heights match, those two cylinders are the same cylinder: radius r, height r.

10:04And two thirds plus one third is one whole. So the toy occupies exactly pi r cubed, which is one such cylinder, entire. The enclosing cylinder has radius r and height r plus r, so it occupies 2 pi r cubed. The toy is precisely half of it, at every radius there is, and the leftover has no choice but to equal the toy. It is the matching that does the work, not the shapes.

10:35Make the cone twice as tall as the radius and the toy fills four ninths of its cylinder. Half as tall, and it fills five ninths. Only when the height equals the radius do the two thirds and the one third land on one whole. Here is that solid again with the radius taken as 1, and the cone 1 tall to match. You could grind through it: two thirds of pi, plus one third of pi.

11:05Or notice it is the identity you have just proved, at r equal to 1. Either way the answer is pi cubic centimetres. Exactly pi. Not 3.14, and not 3.1416. When you are asked for an answer in terms of pi, that exact form is the answer, and it is the form that shows the structure. 3.14 is a stand-in you substitute at the very end, and only if somebody wants a decimal.

11:40Put it in too early and you lose the one thing worth seeing. The rule was never about two pieces. Here are three. A model made of a cylinder with a cone at each end, 12 centimetres long overall, 3 across. The radius is 1.5, and the cones take 2 centimetres each, so the cylinder is 12 less 4, which is 8 long. The cylinder measures 18 pi, each cone 1.5 pi, and the three add to 21 pi, or 66 cubic centimetres.

12:17Next, a sweet shaped like a cylinder with a rounded cap at each end. 5 centimetres long, 2.8 across. The radius is 1.4, so the caps take 2.8 between them and the cylinder is 2.2 long. Two hemispheres of one radius make a whole sphere, which is a tidy way to add them, and one sweet comes to about 25.05. 45 of them is about 1127.3, and syrup filling 30 per cent of that is roughly 338 cubic centimetres.

12:58Last, an iron pole: a cylinder 220 centimetres tall of base diameter 24, carrying a second one 60 tall of radius 8. Read that again. The first is given by its diameter and the second by its radius, and that is where marks go missing. 99475.2 and 12057.6 make 111532.8 cubic centimetres, and at 8 grams for each of those, about 892 kilograms. So there was never a second rule.

13:43There is one observation, and it is about what a face is. A face is flat. It has an area, and it has no thickness, so it takes up no room. Join two solids along one and the room they occupy is simply the two rooms put together. Nothing is shared, so nothing is double counted, and nothing is lost. The boundary is the opposite story, for the same reason: that face was on both outsides, and now it is on neither.

14:16Volumes add. Surface areas fall short, by twice the contact. Both are one fact, read off two different measurements. Learn the fact, and you never have to remember which way round the rules go.