Composite Solids

Some solids do not match any formula you know, and this one is a good example: a step shape, 10 centimetres wide, 4 deep and 6 tall at its highest point. Stop hunting for a single formula, because there is not one. Instead notice what it really is — a composite solid, built out of simpler solids joined together, or with a piece taken away. Break it into rectangular prisms, deal with each prism on its own, then combine the results. There are two ways to do that combining. You can add, by cutting the solid into two prisms and adding their volumes. Or you can subtract, by taking the full box and removing the gap. Both give the same answer.

The adding method first. Cut the step shape straight across, 3 centimetres up from the base, and you are left with two ordinary rectangular prisms. The bottom piece runs the full width: 10 centimetres wide, 4 deep and 3 tall, so its volume is 10 times 4 times 3, which is 120 cubic centimetres. The top piece is only the tall part: 6 centimetres wide, 4 deep and 3 tall, giving 72 cubic centimetres. Add them and the whole solid is 192 cubic centimetres. The thing that catches people is measuring the pieces. After you cut, each piece has its own sizes — the top piece is 6 wide, not 10, and each piece is 3 tall, not 6.

Now the same solid the other way round. Pretend the step is not there and work out the full box: 10 wide, 4 deep and 6 tall gives 240 cubic centimetres. Then work out the piece that is missing. The gap is a rectangular prism as well — 4 centimetres wide, because 10 take away 6 is 4; 4 deep, the same as everything else; and 3 tall, because 6 take away 3 is 3. That gap is 4 times 4 times 3, which is 48 cubic centimetres. Subtract: 240 take away 48 gives 192 cubic centimetres, exactly what the adding method gave. Since the two methods are independent, doing both is a genuine check on your answer.

Not every composite solid is a step. Plenty are simply one prism stacked on another, like a block sitting on a slab, and those are always an adding job. The slab here is 12 centimetres by 8 by 2, so its volume is 192 cubic centimetres — a big number from a small height, because it is wide and flat. The block on top is 4 by 4 by 5, giving 80. Add them for 272 cubic centimetres. The one thing to guard against is borrowing measurements between the pieces. The slab is 8 centimetres deep but the block is only 4 deep, and using 8 for both would add 80 cubic centimetres of solid that simply is not there.

Here is the routine, five steps every time. Split the solid into two rectangular prisms, or spot that it is a box with a gap in it. Write down each piece's own length, depth and height, working out any missing side by subtracting. Find each volume separately with length times depth times height. Then add the pieces together, or take the gap away from the full box. Finally check using the other method and put cubed units on your answer. As a last example, a box 8 by 5 by 6 is 240 cubic centimetres, and cutting a corner of 3 by 5 by 2, which is 30, leaves 210. This is real work: concrete for a stepped path, soil for a raised garden bed, packing an odd-shaped parcel.