Finding a cuboid dimension from its volume

A tank has a square base. Here's the tank. Its volume is nine hundred and seventy two cubic centimetres and its height is 12 centimetres. Find the length of one side of the base. The base has no numbers on it at all. The volume of a box is its length, times its breadth, times its height. This tank gives us the volume, and it gives us the height. It doesn't give the length, and it doesn't give the breadth. Cut off one layer, one centimetre thick. The top of that layer is the floor of the tank. That floor is the base area. So the length times the breadth is the base area, and the rule reads: base area, times height. The base area is the one thing this question doesn't give. So work it back: the volume, divided by the height. Nine hundred and seventy two divided by 12 is 81. And cubic centimetres, divided by centimetres, leaves square centimetres. The other way round, a volume divided by its base area leaves the height. The base is a square. All four sides of a square are the same, so this side and this side are the same number. That number, times itself, gives 81. The number that does that is called the square root of 81. 9 times 9 is 81, so one side of the base is 9 centimetres.

Your turn. A solid metal cube has a volume of five hundred and twelve cubic centimetres. Find the length of one edge. A cube's edges are all the same. Try it first. Pause if you like. Same rule, and the same first line. Length, times breadth, times height. In a cube, all three are the same number, so the volume is one edge, times itself, times itself. 8 times 8 times 8 is five hundred and twelve. So one edge is the cube root of five hundred and twelve. It is 8 centimetres. A cube is a floor of squares, stacked. The side of a square is the square root of its area. The edge of a cube is the cube root of its volume.