the area of a rectangle
How much paint covers a wall, how much carpet covers a floor, how many tiles fill a courtyard — these all ask for area, the amount of flat surface a figure covers. For a rectangle the answer is wonderfully simple, and it is the seed from which almost every other area formula grows.
Tile the rectangle with unit squares. A rectangle that is b units wide and h units tall holds exactly b rows of h squares, or h columns of b squares — either way b times h little squares. So the area is A = b · h, base times height. A 3-by-5 rectangle is covered by 15 unit squares, so its area is 15 square units. For a square, base and height are equal, giving A = s^2.
Notice the units multiply too: metres times metres give square metres (m^2), which is why area is a 'squared' quantity. This counting-of-squares picture is exactly why area scales as the square of a length: doubling both sides of a rectangle quadruples its area, the seed of the square-cube law.
A room measuring 4 m by 3 m has floor area 4 × 3 = 12 m^2, so 12 square metres of carpet will cover it exactly.
Area = base × height; the units multiply to square units.
Base and height must be measured in the same unit before multiplying; mixing metres and centimetres silently corrupts the answer.