Pythagorean Theorem: Visual Proof

The Pythagorean theorem states that for a right triangle with legs $a$ and $b$ and hypotenuse $c$:

$a^2 + b^2 = c^2$

Proof by area rearrangement

Consider a large square with side length $(a + b)$. Its total area is $(a + b)^2$.

Place four identical right triangles (each with area $\frac{ab}{2}$) inside this square. The remaining area equals:

$(a + b)^2 - 4 \cdot \frac{ab}{2} = a^2 + 2ab + b^2 - 2ab = a^2 + b^2$

In the first configuration, the remaining area forms a single square with side $c$, so the remaining area is $c^2$.

In the second configuration, the remaining area forms two squares with sides $a$ and $b$, so the remaining area is $a^2 + b^2$.

Since both equal the same remaining area: $a^2 + b^2 = c^2$. $\blacksquare$

Click the visualization to toggle between configurations: