One equation, four civilisations, and hundreds of proofs. In every right triangle, a² + b² = c² — and c is always the side opposite the right angle. Never “the last number given”.
📐Name the Parts Before You Compute
The two sides forming the right angle are the legs (a and b — interchangeable). The side directly opposite the right angle is the hypotenuse (c), and it is always the longest. Getting c right is the single most important step in any Pythagorean problem.
🧩Why It Is True: Pack One Square Two Ways
Make four copies of the same right triangle. Draw one big square with side (a + b). Now fill that same square twice, using the same four triangles — and compare what is left over. The only idea you need: sliding a shape does not change its area.
📌The whole argument in one sentenceThe outer square is the same size both times, and the same four triangles were removed both times — so whatever is left must have the same area both times. Left over: c² and a² + b². Therefore they are equal, for every right triangle, with nothing measured anywhere.
🎒Why the tilted region really is a square. In any right triangle the two acute angles add to 90°. Where two hypotenuses meet on an edge, those two acute angles sit beside one corner of the tilted region and together fill a straight line of 180° — leaving exactly 90° for that corner. Four equal sides, four right angles: a square. And Arrangement 1 is the xián tú (hypotenuse figure) from the Chinese classic Zhoubi Suanjing — the picture carried this proof for centuries before anyone spelled it out in sentences.
🔧Two Jobs, One Equation
Same formula, different rearrangement — and mixing them up is the number-one error on this topic. Add to find the hypotenuse. Subtract to find a missing leg.
➕Finding the hypotenuse → ADD
Legs 5 cm and 12 cm. Find c.
5² + 12² = c²
25 + 144 = 169
c = √169 = 13 cm
c came out bigger than both legs — a hypotenuse always must. That is your sanity check.
➖Finding a leg → SUBTRACT
Hypotenuse 25, one leg 7. Find the other.
7² + b² = 25²
49 + b² = 625 → b² = 576
b = √576 = 24
24 is smaller than the hypotenuse 25 ✓. If a “leg” comes out longer than c, you added when you should have subtracted.
A 17-foot ladder leans on a wall, base 8 feet out. How high does it reach?
1
Find the hypotenuse first. The ladder is the slanted side, so c = 17. The wall height and the ground distance are the legs.
2
This is a missing leg problem → subtract: h² = 17² − 8² = 289 − 64 = 225
3
h = √225 = 15 ft. Keep only the positive root — −15 is a perfectly good solution to the equation and a nonsense answer for a length.
4
Sense-check: 15 ft up a wall from a 17 ft ladder, base 8 ft out. The ladder is longer than the height it reaches ✓ and 8-15-17 is a known triple.
⭐ The ladder reaches 15 feet up the wall
🔢Triples Worth Recognising on Sight
A Pythagorean triple is three whole numbers with a² + b² = c². Spot one and you skip the arithmetic entirely. Every whole-number multiple of a triple is also a triple — that is why the 3-4-5 family is everywhere.
Triple
Check it
The family it generates
3 · 4 · 5
9 + 16 = 25 ✓
6-8-10, 9-12-15, 12-16-20, 30-40-50 …
5 · 12 · 13
25 + 144 = 169 ✓
10-24-26, 15-36-39, 25-60-65 …
8 · 15 · 17
64 + 225 = 289 ✓
16-30-34, 24-45-51 …
7 · 24 · 25
49 + 576 = 625 ✓
14-48-50, 21-72-75 …
20 · 21 · 29
400 + 441 = 841 ✓
40-42-58, 60-63-87 …
⚠️A triple is a shortcut, not a law. Sides 4, 5, 6 look friendly and are not a triple: 16 + 25 = 41, but 6² = 36. Always verify before you trust a set of numbers on sight — and never assume the largest number is the hypotenuse unless the triangle is actually right.
🔄Flip It: a Portable Right-Angle Detector
Euclid proved the theorem in Elements I.47 and the converse in the very next proposition, I.48. The converse says: if a² + b² = c² for a triangle’s three sides, then the angle opposite c has to be a right angle. That turns a description into a tool.
🟢a² + b² > c²The angle opposite c is ACUTE. The legs overshoot — the hinge is closed past 90°.
🔵a² + b² = c²Exactly RIGHT. Equality lands at 90° and nowhere else.
🔴a² + b² < c²The angle opposite c is OBTUSE. The legs fall short — the hinge has opened past 90°.
🔍Test: sides 7, 9, 12
c must be the LONGEST side: c = 12.
7² + 9² = 49 + 81 = 130
12² = 144
130 < 144
Not a right triangle — and because the sum falls short, the angle opposite 12 is obtuse.
🧵The rope-stretchers’ corner
A loop with 12 evenly spaced knots folds into 3 + 4 + 5.
3² + 4² = 9 + 16 = 25 = 5²
By the converse, the angle between the 3-section and the 4-section must be 90°.
Framers still do this: 3 ft one way, 4 ft the other, adjust until the diagonal reads exactly 5 ft. Scale to 6-8-10 or 12-16-20 for more precision.
🗺️Drop It on a Grid: the Distance Formula
The distance formula is not a new result. It is the Pythagorean Theorem wearing coordinates: the horizontal and vertical gaps are the legs, and the straight-line distance is the hypotenuse.
🔑Key Terms
📐Right TriangleA triangle containing exactly one 90° angle. The theorem applies to these and only these — no near-misses.
🦵LegEither side forming the right angle. The a and b in a² + b² = c², and it does not matter which you call which.
➡️HypotenuseThe side directly opposite the right angle — always the longest. It is c, and identifying it is the first move in every problem.
🎖TheoremA statement proved by logical argument, not checked on examples. Once proved it holds for every case forever.
➕Pythagorean Theorema² + b² = c². The squares built on the legs have exactly the combined area of the square built on the hypotenuse.
√Square RootThe inverse of squaring. Since 13² = 169, √169 = 13. Keep only the positive root for a length — distances cannot be negative.
🔢Pythagorean TripleThree whole numbers satisfying the equation: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Every whole-number multiple is also a triple.
🔄ConverseSwap the if and the then: if a² + b² = c², the triangle must be right. That makes the theorem a test, not just a formula.
🗺️Distance Formulad = √[(x₂−x₁)² + (y₂−y₁)²] — the theorem written for a coordinate grid, with Δx and Δy as the legs.
📜Plimpton 322A Babylonian clay tablet from around 1800 BCE listing Pythagorean triples — including 65-72-97 (4225 + 5184 = 9409 = 97²). Roughly 1,200 years before Pythagoras.
🌎Where This Shows Up
🔨Construction and carpentry. Framers square up walls and foundations with the 3-4-5 method: 3 feet along one wall, 4 feet along the other, and adjust until the diagonal reads exactly 5 feet. On big jobs they scale to 6-8-10 or 12-16-20 for precision. A corner a quarter-inch off at the foundation becomes a wildly crooked room three storeys up.
📱Your phone, and every screen you own. When an app says a destination is 2.3 miles away, or a game decides whether an enemy is close enough to notice you, some version of √(Δx² + Δy²) is running underneath. A “55-inch” TV is 55 inches on the diagonal — the hypotenuse of the screen rectangle, which is why two 55-inch sets with different aspect ratios have different areas.
📚A proof collection that never stops growing. In 1876 a sitting U.S. congressman, James A. Garfield, published an original proof using the area of a trapezoid; five years later he became president. In 1927 Elisha Scott Loomis collected 367 distinct proofs in one book. And in 2023 two New Orleans high-school seniors, Ne’Kiya Jackson and Calcea Johnson, presented a trigonometric proof — an approach experts had long assumed was circular and impossible. Their work, by then ten proofs, was published in the American Mathematical Monthly in 2024.
🛑The door it cracked open. Build a right triangle with both legs equal to 1, and the hypotenuse is √2 — a number that cannot be written as any fraction of whole numbers. The theorem that looked like a tidy statement about tidy numbers turned out to guarantee the existence of untidy ones.
📌Remember This
1The theorem works only for right triangles, and c is always the hypotenuse — the side opposite the right angle, never just “the last number given”. Identify it before you write anything down.
2To find the hypotenuse, add the squares of the legs. To find a missing leg, subtract the smaller square from the hypotenuse’s square. Same equation, different rearrangement — and mixing them up is the number-one error here.
3The converse flips the theorem into a test. If a² + b² = c², the triangle has a right angle. If a² + b² > c², the angle opposite c is acute; if less, it is obtuse.
🤔 Think about it
Picture two sticks of fixed length a and b hinged at one end, with c stretched between their free ends. Open the hinge past 90° and c grows, so a² + b² falls short. Close it below 90° and a² + b² overshoots. Why does equality land exactly at the right angle and nowhere else?
The 2D distance is √(Δx² + Δy²) and the 3D distance is √(Δx² + Δy² + Δz²). Prove the 3D version by applying the theorem twice — once on the floor of a rectangular box, then again using that floor diagonal as a leg. What would a 4D version look like, and would the same argument still work?
⭐Remember: find the right angle, then find the side across from it — that is c. Add for the hypotenuse, subtract for a leg, and let the answer earn its size before you write it down.