Square Area

The square is the unit of area itself. When we say a room is 12 square metres, we mean it would take twelve 1 m × 1 m squares to tile it — so $A = s^2$ is less a formula than a definition made arithmetic. It is also the reason area scales as the square of size: double a square's side and you quadruple its area, which is why a 12-inch pizza is not twice the food of a 6-inch one but four times it.

Going backwards, $s = \sqrt{A}$, answers the tiler's question: 20 m² of tile covers a square room 4.47 m on a side. The square root is where irrational numbers were born — the Pythagoreans discovered that a square of area 2 has a side that no fraction can express, a fact that reportedly disturbed them enough to keep it secret.