A proof of the Pythagorean theorem is an argument showing that a2 + b2 = c2 holds for every right triangle, where a and b are the legs and c is the hypotenuse. Measuring triangles can't do that; a proof can. Hundreds of proofs are known. Below is one you can watch, then a map of the twelve on this site.
A proof you can watch
Both big squares have side a + b, so they have the same area. Each holds four identical copies of the right triangle. On the left, the uncovered region is a tilted square with side c. On the right, the triangles have slid together into two rectangles, and the uncovered region is a square of side a plus a square of side b.
Equal squares minus equal triangles leave equal areas, so c2 = a2 + b2. Nothing in the argument depends on the particular lengths, so it works for every right triangle. The full version, including why the tilted shape really is a square, is on the rearrangement proof page.
A proof versus checking examples
You can test the theorem on a 3-4-5 triangle by counting grid squares: 9 + 16 = 25. That is a useful check, but it isn't a proof.
Three things go wrong with "it worked every time I tried it":
- You can't try every triangle. There are infinitely many shapes of right triangle, and most have sides like √7 that no grid can count.
- Measurements are approximate. A ruler can't tell 5 from 5.001, so measuring can never show that two areas are exactly equal.
- Patterns can fool you. The expression n2 + n + 41 gives a prime number for n = 0, 1, 2, …, 39, forty successes in a row, and then fails at n = 40, where it equals 1681 = 412.
A proof avoids all three. It starts from facts already established, such as "the angles of a triangle add to 180°", "congruent shapes have equal areas" and "a square of side s has area s2", and uses logic to reach the conclusion for all right triangles at once. Examples still matter: they build intuition and catch mistakes. They just can't finish the job.
NoteThe theorem and its converse are different statements, and each needs its own proof. The theorem says every right triangle satisfies a2 + b2 = c2; the converse says every triangle that satisfies it is a right triangle.
Every proof on this site
Proofs of the theorem fall into families, named after the tool that does the work. Here are the twelve explained on this site, roughly from the most visual to the most abstract.
| Proof | Type | Key idea | Level |
|---|---|---|---|
| Rearrangement | Dissection | The same four triangles leave c2 in one (a + b)-square and a2 + b2 in another | Start here |
| Chinese (gougu) | Dissection | Pieces move "out and in" on the hypotenuse diagram | Beginner |
| Bhāskara | Dissection | Four triangles and a small square make c2, then a2 + b2 | Beginner |
| Algebraic | Area + algebra | (a + b)2 = 4 · 1 over 2ab + c2, then expand | Beginner |
| Garfield | Area + algebra | The area of a trapezoid computed two ways | Intermediate |
| Perigal | Dissection | Four pieces of the b2 square fit round a2 to make c2 | Intermediate |
| Similar triangles | Similarity | The altitude splits the triangle into two smaller copies of itself | Intermediate |
| Einstein | Similarity + area | Areas of similar triangles grow with the square of the hypotenuse | Intermediate |
| Euclid (Elements I.47) | Area + congruence | Shear each leg square into one part of the hypotenuse square | Advanced |
| Leonardo da Vinci | Area + symmetry | Two congruent hexagons built from copies of the triangle | Advanced |
| Ptolemy's theorem | Circle geometry | Apply Ptolemy's theorem to a rectangle drawn in a circle | Advanced |
| Trigonometric | Trigonometry | Use trigonometry without assuming the theorem along the way | Advanced |
The families differ in what they rely on. Dissection proofs cut shapes into congruent pieces and rearrange them, so only "moving a piece doesn't change its area" is needed. Area-and-algebra proofs write one area in two ways and simplify. Similarity proofs compare the side ratios of similar triangles and can avoid area altogether. Trigonometric proofs need extra care, because the identity sin2θ + cos2θ = 1 is itself the theorem in disguise (see the Pythagorean identity), and using it would be circular.
How many proofs are there?
Nobody knows exactly, because deciding when two proofs are "different" is a judgment call. The best-known collection is The Pythagorean Proposition by the American mathematics teacher Elisha Scott Loomis (1852–1940). Its enlarged second edition of 1940 is usually described as containing 367 proofs, although published counts of the same book range from 344 to 370 depending on what is treated as distinct. The National Council of Teachers of Mathematics reissued it in 1968.
Loomis sorted most of his proofs into algebraic ones, based on relations between lengths, and geometric ones, based on comparing areas. He also asserted that no trigonometric proof is possible, because the basic formulas of trigonometry already depend on the theorem. Trigonometric proofs that avoid that trap have since been published, including a 2024 paper by Ne'Kiya Jackson and Calcea Johnson, whose story is on the trigonometric proof page.
The oldest proof that survives in writing is Proposition 47 of Book I of Euclid's Elements, from around 300 BCE. The rule itself is far older: Babylonian scribes were working with Pythagorean triples around 1800 BCE, as the history of the theorem explains.
Which proof should you learn first?
- The rearrangement proof. No algebra and no formulas, and you can build it from paper. If you learn only one proof, make it this one.
- The algebraic proof. The same picture written as an equation. It is the version most often asked for in homework.
- The similar triangles proof. The standard proof in high-school geometry and the shortest to write down. It also leads to the geometric mean theorem.
- Euclid's proof. The classic, and a real workout in congruent triangles once you are comfortable writing geometric proofs.
Teachers looking for a hook can open with President Garfield's proof, written by a congressman five years before he became president, or with the Chinese hypotenuse diagram, which shows that the idea belongs to more than one culture. Classroom versions of several proofs are on the activities page.
All proof pages
- ProofsThe Rearrangement Proof of the Pythagorean TheoremThe rearrangement proof of the Pythagorean theorem, animated: four triangles in two squares, why the middle is a square, a full write-up and a paper activity.
- ProofsAlgebraic Proof of the Pythagorean TheoremThe algebraic proof of the Pythagorean theorem: (a + b)² = 4·½ab + c² expanded, the (b − a)² version, why the middle is a square, and a two-column proof.
- ProofsEuclid's Proof of the Pythagorean TheoremEuclid's windmill proof of the Pythagorean theorem, Book I Proposition 47, step by step: congruent triangles, an animated shear and the converse, I.48.
- ProofsSimilar Triangles Proof of the Pythagorean TheoremProve the Pythagorean theorem with similar triangles: the altitude to the hypotenuse makes three similar right triangles, so a² = cp, b² = cq and a² + b² = c².
- ProofsBhāskara's Proof of the Pythagorean TheoremBhaskara's "Behold" proof of the Pythagorean theorem, animated: the square on the hypotenuse becomes a² + b², plus the algebra and his second, similarity proof.
- ProofsPresident Garfield's Trapezoid ProofJames Garfield's 1876 trapezoid proof of the Pythagorean theorem: an animated build, the area found two ways, why angle ABD is 90°, and the story behind it.
- ProofsThe Chinese Proof: Gougu and the Hypotenuse DiagramThe Chinese proof of the Pythagorean theorem: the xian tu hypotenuse diagram from the Zhoubi Suanjing, the 3-4-5 grid and Liu Hui's out-in principle, animated.
- ProofsPerigal's Dissection Proof of the Pythagorean TheoremHenry Perigal's dissection proof of the Pythagorean theorem: cut the larger leg square into four equal pieces, then slide them and the small square into c².
- ProofsLeonardo da Vinci's Proof of the Pythagorean TheoremThe hexagon proof of the Pythagorean theorem credited to Leonardo da Vinci: two equal hexagons, one quarter turn, the full proof and doubts about who found it.
- ProofsEinstein's Proof of the Pythagorean TheoremThe similar-triangle area argument linked to Albert Einstein's boyhood proof of the Pythagorean theorem, step by step, plus the dimensional-analysis version.
- ProofsTrigonometric Proofs of the Pythagorean TheoremCan trigonometry prove the Pythagorean theorem without circular reasoning? Jason Zimba's 2009 proof and the Jackson–Johnson law-of-sines proof, explained.
- ProofsProving the Pythagorean Theorem with Ptolemy's TheoremHow Ptolemy's theorem for cyclic quadrilaterals, AC·BD = AB·CD + AD·BC, proves a² + b² = c² in one line, plus a proof sketch and why it is not circular.
Frequently asked questions
How many proofs of the Pythagorean theorem are there?
Several hundred have been published, and new ones still appear. Elisha Scott Loomis's 1940 collection is usually quoted as having 367, but there is no official count, because whether two proofs are "different" is a matter of opinion.
What is the easiest proof of the Pythagorean theorem?
For most people it is the rearrangement proof. It needs only two facts: the area of a square is its side squared, and sliding a shape doesn't change its area. The algebraic proof is the easiest to write out as a formal proof.
Who first proved the Pythagorean theorem?
No one knows. Pythagoras (c. 570–c. 495 BCE) is traditionally credited, but nothing he wrote survives, and the rule was used in Babylon more than a thousand years before him. The oldest proof we can read is in Euclid's Elements, around 300 BCE. See who was Pythagoras?
Can you prove the Pythagorean theorem with trigonometry?
Yes, but carefully. Many trigonometric facts, including sin2θ + cos2θ = 1, are usually derived from the theorem, so using them would be circular. Valid trigonometric proofs build what they need from scratch, as the trigonometric proof page explains.
Why do we need to prove it if it is in the textbook?
Because so much is built on it: the distance formula, trigonometry and the geometry inside GPS and computer graphics. A proof guarantees that a2 + b2 = c2 is exactly true for every right triangle, not just approximately true for the ones somebody measured.
Sources and further reading
- Euclid, Elements, Book I, Proposition 47, with commentary by D. E. Joyce: Clark University edition.
- Elisha Scott Loomis, The Pythagorean Proposition, 2nd ed. (1940; reissued by the National Council of Teachers of Mathematics, 1968).
- Eli Maor, The Pythagorean Theorem: A 4,000-Year History (Princeton University Press, 2007).
- Roger B. Nelsen, Proofs Without Words: Exercises in Visual Thinking (Mathematical Association of America, 1993).
- Alexander Bogomolny, Pythagorean Theorem and its many proofs, Cut the Knot.
- Pythagorean theorem, Wikipedia (overview of proofs and history).
Keep exploring
- ProofsThe Rearrangement Proof of the Pythagorean TheoremThe rearrangement proof of the Pythagorean theorem, animated: four triangles in two squares, why the middle is a square, a full write-up and a paper activity.
- ProofsSimilar Triangles Proof of the Pythagorean TheoremProve the Pythagorean theorem with similar triangles: the altitude to the hypotenuse makes three similar right triangles, so a² = cp, b² = cq and a² + b² = c².
- ProofsEuclid's Proof of the Pythagorean TheoremEuclid's windmill proof of the Pythagorean theorem, Book I Proposition 47, step by step: congruent triangles, an animated shear and the converse, I.48.
- HistoryHistory of the Pythagorean TheoremThe history of the Pythagorean theorem from Babylon (1800 BCE) to a 2023 proof: who discovered it, who proved it first, and why it bears Pythagoras' name.
- LearnConverse of the Pythagorean TheoremThe converse of the Pythagorean theorem explained: the statement, a clear proof, how to test any three side lengths, and many yes-or-no worked examples.
- TeachPythagorean Theorem ActivitiesTen hands-on Pythagorean theorem activities and project ideas, from a paper-puzzle proof and 12-knot rope to a treasure hunt, with materials, steps and prompts.