The theorem says the square on the longest side equals the two smaller squares added together: a² + b² = c². Reading it left to right is the whole trick — if you know the two short sides (the legs), you can build the long one (the hypotenuse).
Add the Squares, Then Undo the Square
Say the legs are 6 and 4. Their squares are 36 and 16, and added together that's 52 — so c² = 52. But we want c, not c², so we do the opposite of squaring: a square root. c = √52 ≈ 7.2. In the toy, drag the legs and you'll see that the slanted side is always exactly the square root of the other two squares added up.
Why the Square Root Has to Be There
The theorem is about areas — the actual squares sitting on each side. So a² + b² gives you the area of the big square, which is c². A length isn't an area, though, so to turn that area back into the side length you take its square root. Press “show the squares” and you'll literally see the green square on the hypotenuse matching the brass and mustard ones combined.
So Defining It and Using It Are Different Jobs
Last concept you proved the relationship is true. Here you're running it as a tool: known legs in, hypotenuse out. Once that feels automatic, the very next rung adds the second case — finding a shorter side — which is the same idea run slightly differently.