Why does a² + b² = c²?
Put a square on each side of a right triangle. The two small ones, poured together, fill the big one exactly. Change the sides and it keeps happening.
9 + 16 = 25
So c = 5. The big square is exactly the two small ones together.
- Side a
- 3
- Side b
- 4
- Square on side a
- 9
- Square on side b
- 16
- Square on side c
- 25
- Long side c
- 5
Drag the end of either short side to change its length.
Jump to a triangle
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Where it usually goes wrong
a² + b² = c² gets used on any triangle with three numbers on it, and c gets taken to be whichever length was mentioned last. The rule is about right triangles only, and c is always the side opposite the right angle — the longest one. Square the wrong pair and the arithmetic is flawless and the answer is nonsense.
Try this
- Start at 3-4-5, then drag a short side. The two small squares always pour into the big one exactly, whatever numbers you land on.
- Try 6-8-10. It is the 3-4-5 triangle at double size, which is why the same three squares fit.
- Press 2-3-? and read the long side. Most right triangles do not have a whole number there — 3-4-5 is the exception people remember, not the rule.
Where this goes next
This is how any distance gets measured on a grid — on a map, on a screen, between two points in a coordinate plane. Distance Between Points is the same triangle with the squares taken away.