The volume
Take the box apart into layers and lay them side by side.
- Length
- 6 cm
- Width
- 4 cm
- The height
- 3 cm
- One layer
- 24 cm²
- Number of layers
- 3
- Volume
- 72 cm³
Regnestykket
Length × Width × Height
6 cm × 4 cm × 3 cm = 72 cm³
Card: l · b · h
Nothing disappeared
The box didn't get smaller by being taken apart. The slices lying in a row are exactly the same as the ones that lay on top of each other — there are just as many, they are the same size, and they have the same shape. That is why there is as much room in the row as in the box. It is the same idea as in the rectangle, where the squares were laid out on a line: you move something to where it is easier to count, without changing how much there is.
One layer is just a rectangle
Look at a single slice. On top it is an ordinary rectangle with a length and a width, and its area is the two multiplied together — just as in the rectangle workshop. That number is shown underneath as «one layer». The slice is one centimetre thick, so it takes up as many cubic centimetres as it is square centimetres big. So there is nothing new to learn here: you already know the base area, and you just need to know how many of those lie on top of each other.
The height is the number of layers
Drag the height slider and count the slices in the row. One slice is added for each centimetre. That is why the volume becomes the base area multiplied by the height: you add the same base area once for every centimetre upwards. Multiplying is adding equal things together many times, and here you can see exactly what those equal things are. Set the height to 2 and notice that the volume is twice as big as one layer.
A cubic centimetre is a cube
The area was measured in square centimetres — small squares one centimetre by one centimetre. Volume is measured in cubic centimetres, a small cube one by one by one centimetre, and it is written cm³. The small three means that three measurements are multiplied together instead of two. That is why you can't compare an area and a volume: one counts surfaces, the other counts blocks. A litre is a thousand cubic centimetres, that is a cube ten by ten by ten.
The cylinder is the same rule
A tin, a pipe or a tank isn't a box — but it is still a stack of layers. Each layer is a circle instead of a rectangle, and you can find the circle's area in the circle workshop. After that the calculation is the same: the base area times the height. That applies to anything with the same cross-section all the way up, whatever shape the cross-section has. That is why it's worth getting to know one layer properly — the rest is just multiplying by how tall it is.
Where you'll meet it again
Concrete to be ordered for a foundation, insulation to fill a cavity, water in a hot-water tank, air in a ventilation duct, gravel on a driveway, waste water in a tank. Each time the question is the same: how much room is there. And each time the answer is the base area times the height. If the space is skewed or has a corner cut off, you split it into boxes and prisms and add them up — just as you split a skewed surface into rectangles and triangles.