The rectangle
Cut it into squares and lay them in a row. Now you can count it.
- Length
- 6 cm
- Width
- 4 cm
- All the way round
- 20 cm
- Regnestykket
- 6 × 4
- Number of bays
- 24
- Area
- 24 cm²
Regnestykket
Length × Width
6 cm × 4 cm = 24 cm²
Card: l · b
The row is the same as the rectangle
Not a single square was added, and not a single one disappeared. The squares were just moved out onto one line. So there is just as much room in the long row as in the rectangle you started with — it is the same figure, cut up and laid out differently. And a row is easy to count. That is the whole idea: a rectangle is hard to take in at once, but a row is something you can go through from one end to the other.
One square is a square centimetre
Each of the small squares is one centimetre long and one centimetre wide. Such a square is called a square centimetre, and it is written cm². That is what you count when you say how big something is. That is why the same number is shown next to number of squares and next to the area: if you have counted the squares, you have found the area. Not because someone decided it should match, but because the area IS the number of squares.
The two copper lines measure along
One copper line lies along the length, the other stands along the width. Drag the slider to unfold, and watch them along the way: the length line ends up under the first group in the row, and it is exactly as long as before. The width line lies down across the whole row. Where it counted the rows in the grid, it now spans just as many groups. So the lines measure the same thing all the way — they just point to it in a new place.
That's why it's times, not plus
Look at the gaps in the row. They divide it into equal groups, and there is one group for each row in the grid. All the groups are the same length, because all the rows in the grid were the same length. Adding up equal groups is exactly what times means. So instead of counting the squares one at a time, you can take the length and multiply it by the width. Try setting the width to 1: then there is only one group, and the row is exactly as long as the rectangle is wide. Turn the width up one step at a time, and see how one more group is added each time.
A square is not a different kind of figure
Set the length and the width to the same number. Now the figure is a square. But nothing special happened along the way — it is still the same rectangle, and it is still worked out the same way: length times width. A square is simply the rectangle where the two measurements happen to be the same. That is why there is no extra formula to learn for squares, and that is why all squares are rectangles, while very few rectangles are squares.
All the way round is something quite different
Try a rectangle of 1 by 7 and then one of 4 by 4. Both are 16 centimetres all the way round — it is exactly as far to walk along their edges. But the first has 7 squares in it, and the second has 16. That is more than twice as much room for the same edge. So the length of the edge doesn't tell you how much is inside. It is the same trap as in the triangle, where the sides get longer without the area changing, and it is worth having seen it with your own eyes at least once.
Where you'll meet it again
A floor that needs boards, a wall that needs painting, a sheet to be cut to size, a piece of insulation on a surface. Each time the question is the same: how many square centimetres, square metres or whole sheets is there room for. And each time the answer is the same calculation as here — measure one way, measure the other way, multiply them. If the figure becomes skewed or gets a corner cut off, you split it into rectangles and triangles and add them up. That is why the first two shapes are the ones everything else builds on.