Pythagoras by rearrangement

Where’s the maths? This is geometry — the part of it called mensuration, where a length is turned into an area. Every right-angled triangle here carries three squares, one built on each side, and the area of a square is simply its side multiplied by itself. Pythagoras’ theorem says the two smaller squares always hold exactly as much as the biggest one. The water wheel makes that literal: turn it half a turn and the liquid that filled the two small chambers fills the big one to the brim, with nothing spare and nothing missing. The dissection tabs make it visual instead — cut the squares into pieces and slide the pieces about, and because sliding a shape never changes its area, the equality is proved with no arithmetic at all. Algebra joins in as well, because expanding (a + b)² = a² + 2ab + b² is the single line that finishes the four-triangle proof. And the theorem runs backwards too: if three lengths satisfy a² + b² = c², the triangle they make must be right-angled — which is how a builder checks that a corner is truly square.

The water wheel. This is the classic science-museum machine. The triangle and its three squares are sealed, shallow chambers on one turning disc, joined to each other by small channels at the triangle’s corners, so the whole thing is a single vessel. It holds exactly a² + b² of liquid. Start with the big square at the top: the two small chambers are brim full. Turn the wheel half a turn and every drop drains into the big chamber — which it fills exactly, to the rim, with the two small ones bone dry. The liquid is the proof: the same amount fits both ways round, so a² + b² = c². In between, the liquid always finds one level, and the three fill percentages add up to that same total.

The (a + b)² proof. Four copies of the triangle sit inside a square of side a + b. Arranged as a pinwheel they leave a tilted square hole of side c; slid into two rectangles they leave holes of a² and b². The four triangles never change, so the two holes are equal: c² = (a + b)² − 4 × ½ab = a² + 2ab + b² − 2ab = a² + b².

Perigal’s dissection (1830). Cut the larger of the two small squares with two lines through its centre, one parallel to the hypotenuse and one at right angles to it. That makes four congruent four-sided pieces which, together with the smaller square left whole, slide — no turning, no flipping — straight into the square on the hypotenuse. Five pieces, area a² + b², filling c² exactly.

Similar triangles. Drop a perpendicular from the right angle onto the hypotenuse; it splits the triangle into two smaller ones, each the same shape as the original. Matching the sides gives b² = c × q and a² = c × p, so a² + b² = c(p + q) = c², because p and q are the two parts of c.

The converse. Reverse it and you get a test: 5, 12, 13 gives 25 + 144 = 169, so that triangle has a right angle. 5, 6, 7 gives 25 + 36 = 61 while 7² = 49, and since 49 < 61 the biggest angle is acute, not right.

A note on the model. The wheel is idealised: the liquid is allowed to share one level right through the corner channels at every angle. A real wheel briefly traps a pocket of air in a chamber whose channel has risen above the liquid, so its needle lags a little; those trapped-pocket moments are ignored here. The connecting triangle is treated as a duct of negligible volume, so the liquid always accounts for exactly a² + b².

Drag the disc to turn the wheel · drag the background to orbit · scroll or pinch to zoom · drag the gold corner to reshape the triangle · Flip, Spin and the Turn slider drive the wheel.