A square metre is a square that's 1 m by 1 m. We want to know how many square centimetres fit inside it. The catch is that area lives in two directions at once, and that changes everything.
The Guess Everyone Makes
One metre is 100 centimetres — true. So it feels like a square metre should be 100 square centimetres. But that's only counting one row of little squares along the bottom. A square has rows going up as well. Each row is 100 squares across, and there are 100 rows stacked up. Press "fill the whole square" and you'll see the rows pile up one after another until the whole thing's packed.
It Happens Because Area Is 2-D
Length is one direction, so converting length is one multiplication: 1 m = 100 cm, done. Area is two directions, so converting area squares that factor. The metre became 100 centimetres going across, and it became 100 centimetres going down too — 100 × 100. That's why the jump is so much bigger than people expect.
The Same Story for Every Pair
It's not special to metres and centimetres. 1 cm² is a 10 mm by 10 mm square = 100 mm². A hectare is a 100 m by 100 m square = 10 000 m². Same idea every time: take the length factor, then square it because you're filling a 2-D space.