Dragging is checking: however many shapes you try, you have only tried those. Each picture is a proof because its argument never uses your numbers. Every number on the page is computed from the shapes as you move them.
- Rearrange
- Four triangles in one box. What they leave uncovered cannot change: c² before, a² + b² after.
- Swing
- c² is four triangles and a small square. Swing two triangles and the same pieces read a² + b².
- Cut and tile
- Cut the bigger square through its centre. Each cut is c long and the hole is the smaller square, so the two squares tile c².
- Scale
- The drop line makes two smaller copies of the triangle. Their sides give a²/c and b²/c, which add up to c.
- Unroll a circle
- Thin wedges, laid head to tail, make a near-rectangle πr wide and r tall. Polygons inside and outside trap the circle, so its area is πr².
- Staircases
- Two copies of 1 + 2 + … + n make an n × (n + 1) rectangle, so one is half of it.
- Odd numbers
- Each L-shaped corner of 2n − 1 tiles turns an (n − 1) square into an n square.
- Corners
- Spin two corners about the middles of their sides and the three fill a straight line: 180° on a flat page.
In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c².
Euclid, Elements I.47 (D. E. Joyce ed.)
If a² + b² = c² for a triangle's sides, the angle opposite c is a right angle.
Euclid, Elements I.48
For any triangle, c² = a² + b² − 2ab·cos C. When C is 90° the last term is zero; when C is wider, c² is bigger than a² + b²; when it is narrower, smaller.
Euclid, Elements II.12–II.13 (geometric form)
Any similar shapes on the three sides work, not only squares: the shape on the hypotenuse equals the two on the legs.
Euclid, Elements VI.31
The three angles of a flat triangle add to two right angles, 180°.
Euclid, Elements I.32
The proof uses a line through one corner parallel to the opposite side, so it rests on Euclid's parallel postulate. On a sphere the angles add to more than 180°.
Euclid, Elements I.29 and I.32 (Joyce)
On a sphere a triangle's angles add to more than 180°; the excess equals the triangle's area divided by R².
MathWorld, Spherical Excess
A circle's area equals a right triangle with legs r and 2πr: ½ · r · 2πr = πr².
Archimedes, Measurement of a Circle, Prop. 1
Squeezing the circle between 96-sided polygons, Archimedes showed 3 10/71 < π < 3 1/7.
Archimedes, Measurement of a Circle, Prop. 3
1 + 2 + … + n = n(n + 1)/2. Two copies of the staircase make an n × (n + 1) rectangle.
OEIS A000217, triangular numbers
1 + 3 + 5 + … + (2n − 1) = n². Each odd number is an L-shaped corner that wraps one square into the next.
OEIS A000290, the squares
For whole numbers m > n > 0, (m² − n², 2mn, m² + n²) is always a Pythagorean triple. It is primitive (no common factor) exactly when m and n share no factor and are not both odd, and every primitive triple comes from one such pair. Every other triple is a primitive one times a whole number k: (9, 12, 15) is 3 × (3, 4, 5).
Wikipedia, Pythagorean triple: Generating a triple (Euclid's formula, Elements X Lemma 1 before Prop. 29)
In a right triangle, the altitude from the right angle splits it into two triangles with the same angles as the whole (so the same shape, scaled).
Euclid, Elements VI.8
The straight-line distance from (0, 0) to (x, y) is √(x² + y²): the steps across and up are the legs of a right triangle.
Wikipedia, Euclidean distance
Between any two points the distance is √((x₂ − x₁)² + (y₂ − y₁)²): the steps across and up between them are the legs of a right triangle.
Wikipedia, Euclidean distance
In a right triangle the side opposite the right angle is the hypotenuse. It is always the longest side.
Wikipedia, Hypotenuse
Euclid's own proof (Elements I.47) draws this same line from the right angle across the big square, splitting c² into two rectangles equal to a² and b². He shows the areas match with congruent triangles rather than similar ones.
Euclid, Elements I.47 (D. E. Joyce ed.)
Multiply a whole-number right triangle by any whole number and you get another (6, 8, 10 is 3, 4, 5 doubled). Triples with no common factor are called primitive.
Wikipedia, Pythagorean triple
Some staircases are also squares: 1, 36, 1225, 41616, … are both triangular and square numbers, and there are infinitely many.
OEIS A001110, square triangular numbers
√2 is not a fraction. Suppose √2 = p/q with the fraction in lowest terms. Then p² = 2q², so p² is even and so is p. Write p = 2k: then 4k² = 2q², so q² = 2k² and q is even too. Both even contradicts lowest terms.
Wikipedia, Square root of 2: Proofs of irrationality
The length of a curve is defined as the limit of the lengths of polygons drawn along it as their sides shrink. So saying the polygons' perimeters close in on the circumference is a definition at work, and the closing in is a limit.
Wikipedia, Arc length
- c. 1800 BCE
Babylonian scribes were recording Pythagorean triples on a clay tablet around 1800 BCE (Plimpton 322), more than a thousand years before Pythagoras. Each row gives the short side and the diagonal of a right triangle, such as 119 and 169; the long side (120) is not written. Its purpose is debated.
Plimpton 322
- c. 1800 BCE
A Babylonian tablet from about 1800 BCE (Plimpton 322) records much bigger triangles: its first row gives a short side of 119 and a diagonal of 169 (the long side, 120, is left for the reader).
Plimpton 322
- c. 570–495 BCE
Pythagoras (c. 570 – c. 495 BCE) left no writings, and no surviving Greek text from the five centuries after him credits him with the theorem. Who first proved it is unknown.
Wikipedia, Pythagorean theorem: History
- date unknown
The Pythagoreans discovered that a square's diagonal and its side have no common measure: √2 is not a fraction. Little is known for certain about when or how; the story that Hippasus was drowned for revealing it is a legend with little evidence.
Wikipedia, Square root of 2: History
- c. 300 BCE
Euclid's Elements (c. 300 BCE) holds the oldest surviving axiomatic proof (Book I, Proposition 47).
Wikipedia, Pythagorean theorem
- date uncertain
The hypotenuse diagram (xian tu, also romanised xuan tu) appears in the Chinese Zhoubi Suanjing, drawn with a 3-4-5 triangle. The text's date is uncertain.
Xuan tu
- 12th century
Bhāskara II (12th century) drew the same diagram. Legend has it his whole proof was one word: “Behold!”
Xuan tu
- 1575
Francesco Maurolico's Arithmeticorum libri duo (printed 1575) proves the odd-number rule step by step from each square to the next, an argument often called an early proof by induction; how explicit his induction is, historians debate. The L-shaped picture is traditionally linked to the Pythagoreans.
Mathematical induction: History
- 1872 or 1891
Henry Perigal's five-piece dissection is usually dated to 1872, when it appeared in the Messenger of Mathematics; it is certainly in his 1891 booklet Geometric Dissections and Transpositions, and he had it printed on his business cards. When he first found it is uncertain, and sources disagree on the dates.
Cut the Knot, Perigal's proofWikipedia, Henry Perigal
- 1940
There are hundreds of published proofs; one famous 1940 collection (Loomis) lists about 370.
Wikipedia, Pythagorean theorem
Archimedes did not unroll wedges. He compared the circle with a right triangle whose two short sides are the radius r and the circumference 2πr, so its area is ½ · r · 2πr. Trapping the circle between polygons inside and outside it, he showed the circle's area can be neither smaller nor bigger than that triangle's, a method now called exhaustion.
Measurement of a Circle (Wikipedia)
A famous (and probably embellished) story says the young Gauss summed a long arithmetic series in seconds. The oldest account does not say 1 to 100.
Brian Hayes, Gauss's Day of Reckoning, American Scientist 94(3), 2006