Calculating Volume of a Rectangular Prism

You already know that area is how many squares cover a flat shape. Volume is the same idea one step further: how many cubes fill a solid shape. Here is a single layer of centimetre cubes, 4 across and 3 deep. That is 12 cubes in the layer — and notice you found that by multiplying, exactly as you did for area. A layer is just an area made of cubes instead of squares.

Now stack a second layer on top. Each layer holds 12 cubes, and there are 2 layers, so there are 24 cubes altogether. That is the volume: 24 cubic centimetres. You did not count to 24 one cube at a time — you worked out one layer and then multiplied by the number of layers. That is exactly how the formula works, and you have just built it yourself.

So the rule is volume equals length times width times height. For this prism: 4 long, 3 deep, 2 high gives 4 times 3 times 2, which is 24 cubic centimetres. The order does not matter — you can multiply the three numbers in any order you like and get the same answer, so start with whichever pair is easiest. Three numbers instead of two is the only real difference from area.

A cube is the easiest prism, because all three sides are equal. A cube with 3 centimetre sides has a volume of 3 times 3 times 3, which is 27 cubic centimetres. Watch that carefully: it is three multiplications of the side, not two. Stopping at 3 times 3 gives you 9, which is the area of one face, not the volume of the cube. Three dimensions, three numbers.

Here is the link that makes volume genuinely useful. One cubic centimetre of space holds exactly one millilitre of liquid. So a box that measures 10 by 10 by 10 centimetres has a volume of 1000 cubic centimetres, and it holds 1000 millilitres — one whole litre. Volume is the space inside; capacity is how much liquid that space can hold. They are two ways of describing the same thing, and the numbers line up perfectly.