Dissection
Two squares, four triangles, and no algebra
The Pythagorean theorem is usually met as a formula to be memorised. It is much better met as a rearrangement that can be checked by eye.
A circle unrolled into a triangle
Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.
Euclid proves it without moving anything
The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.
Completing the square, by completing a square
The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.
Named alongside it
The objects these essays reach for when they reach for this one.
AreaHypotenuseRight triangleAlgebra tilesAltitudeArchimedesCircle areaCompleting the squareCongruenceCosineImaginary unitIncommensurability