Drag a right triangle and watch area squares animate on each side. Includes algebraic and geometric proofs.
An interactive right triangle whose shape you change by dragging a vertex, with a square rising from each side and its area labeled as you move. Reshape the triangle and the equality a^2 + b^2 = c^2 becomes visible as area: the two smaller squares together exactly fill the largest square. The tool also steps through both the algebraic derivation and the geometric (Euclidean) proof. The thing users most often misread: the numbers on the squares are areas, not lengths, so a leg of 3 carries a square of 9.
A concrete input and expected output from the current implementation.
Input
Drag the triangle until the two legs measure 3 and 4.
Expected output
The leg squares show areas 9 and 16, the hypotenuse reads 5 with square area 25, and 9 + 16 = 25 matches c^2 = a^2 + b^2.
3^2 = 9 and 4^2 = 16, and 9 + 16 = 25, whose square root is 5, so the hypotenuse must be 5. The displayed areas are exactly the three terms of the theorem.
Dragging a vertex changes the two leg lengths, and every side length is squared to size its square and its area label: a side of length 4 carries a square of area 16. Because the figure is a right triangle, the theorem guarantees the hypotenuse square's area equals the sum of the other two.
The algebraic proof builds the big square of side (a + b) around the triangle and subtracts the four identical corner triangles, leaving c^2 = a^2 + b^2. The geometric proof is Euclid's: the two smaller squares are cut into pieces that reassemble, with no gaps or overlaps, into the hypotenuse square, demonstrating the equality by rearrangement instead of algebra.
That is the geometric proof in motion: the pieces of the two leg squares fit together to cover the hypotenuse square exactly, with no gaps and no overlap, which is precisely what a^2 + b^2 = c^2 means as an equality of areas. The proof is complete because the rearrangement works for any right triangle, not just the shape you happen to have on screen.
Only right triangles. For any other triangle the correct statement is the law of cosines, c^2 = a^2 + b^2 - 2ab cos C, and the extra term is zero exactly when C is 90 degrees. So if your shape lacks a right angle, the simple a^2 + b^2 = c^2 equality does not hold, and this tool, which is built around right triangles, will not show it either.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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