The Pythagorean theorem is one of the oldest and most useful mathematical principles ever discovered. It states a simple relationship: in a right triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides. Written as a² + b² = c², this formula has been used for over 4,000 years in construction, navigation, astronomy, and virtually every field that involves measuring distance or angle.
For any right triangle with legs a and b and hypotenuse c:
a² + b² = c²
This formula can be rearranged to find any side. If you know two sides, you can always find the third:
| Find | Formula | Example (a=3, b=4) |
|---|---|---|
| Hypotenuse (c) | c = √(a² + b²) | √(9 + 16) = √25 = 5 |
| Leg (a) | a = √(c² − b²) | √(25 − 16) = √9 = 3 |
| Leg (b) | b = √(c² − a²) | √(25 − 9) = √16 = 4 |
Use the Pythagorean Theorem Calculator to solve for any missing side instantly.
Pythagorean triples are sets of three whole numbers that satisfy a² + b² = c². These are worth memorizing because they appear constantly in math problems, construction measurements, and standardized tests.
| Triple | Verification | Common Multiples |
|---|---|---|
| 3, 4, 5 | 9 + 16 = 25 | 6-8-10, 9-12-15, 12-16-20 |
| 5, 12, 13 | 25 + 144 = 169 | 10-24-26, 15-36-39 |
| 8, 15, 17 | 64 + 225 = 289 | 16-30-34 |
| 7, 24, 25 | 49 + 576 = 625 | 14-48-50 |
Any multiple of a Pythagorean triple is also a Pythagorean triple. The 3-4-5 triple is the most commonly used in construction for checking right angles.
The 3-4-5 rule in construction: To verify a corner is exactly 90 degrees, measure 3 feet along one wall, 4 feet along the adjacent wall, and check that the diagonal between those points is exactly 5 feet. Builders have used this technique for millennia — it requires no specialized equipment, just a tape measure.
The distance formula you learn in algebra — d = √((x&sub2; − x&sub1;)² + (y&sub2; − y&sub1;)²) — is the Pythagorean theorem applied to coordinate geometry. The horizontal distance is one leg, the vertical distance is the other, and the straight-line distance between two points is the hypotenuse. Use the Distance Calculator to compute this for any two points.
This extends to three dimensions: d = √((x&sub2; − x&sub1;)² + (y&sub2; − y&sub1;)² + (z&sub2; − z&sub1;)²). GPS systems, computer graphics engines, and physics simulations all use this 3D extension thousands of times per second to calculate distances between objects in space.
Beyond the 3-4-5 corner check, the Pythagorean theorem is used to calculate rafter lengths for roofs (the run and rise form two legs; the rafter is the hypotenuse), determine diagonal measurements for rectangular rooms (to check if walls are square), find the length of a stairway stringer given the total rise and run, and compute the actual walking distance on a sloped surface. A room that measures 12 feet by 16 feet should have a diagonal of exactly √(144 + 256) = √400 = 20 feet. If the measured diagonal is 19.8 feet, the room is slightly out of square.
Before GPS, sailors used the Pythagorean theorem to calculate the straight-line distance between two points when they could only measure north-south and east-west distances. A ship that sails 30 miles north and then 40 miles east is √(900 + 1600) = 50 miles from its starting point in a straight line. Modern GPS still uses extensions of this principle, calculating distances in three dimensions between satellites and receivers.
Television and monitor sizes are measured diagonally. A “55-inch” TV with a 16:9 aspect ratio has a width of about 47.9 inches and a height of about 27.0 inches. You can verify: √(47.9² + 27.0²) = √(2294 + 729) = √3023 ≈ 55.0 inches. When choosing between screens, the actual viewing area (width × height) matters more than the diagonal number, especially when comparing different aspect ratios.
Baseball uses the Pythagorean theorem implicitly: the bases form a 90-foot square, so the throw from home plate to second base is the diagonal — √(90² + 90²) = √16200 ≈ 127.3 feet. This is why catchers need strong arms for that 127-foot throw. Soccer goalkeepers calculate angles using the same geometry: positioning between the ball and the near post versus far post involves right-triangle relationships.
Although named after the Greek mathematician Pythagoras (circa 570–495 BC), the relationship was known long before his time. Babylonian clay tablets from around 1800 BC contain lists of Pythagorean triples, demonstrating that the concept was understood over a thousand years earlier. Ancient Egyptian rope-stretchers used the 3-4-5 relationship to lay out right angles for building projects. Indian mathematicians documented the theorem in the Sulba Sutras around 800 BC. The Chinese mathematical text Zhoubi Suanjing includes a proof dating to perhaps 1000 BC. Pythagoras (or his school) is credited with providing one of the first formal proofs.
Over 400 distinct proofs of the Pythagorean theorem have been published, including one by President James Garfield in 1876. The theorem has been extended, generalized, and applied in ways that its ancient discoverers could never have imagined, from Einstein’s spacetime calculations to modern machine learning algorithms that compute distance between data points in high-dimensional spaces.
Applying it to non-right triangles. The theorem only works for right triangles (those with one 90-degree angle). For other triangles, you need the law of cosines: c² = a² + b² − 2ab cos(C). Use the Triangle Calculator to solve any triangle type.
Confusing legs with hypotenuse. The hypotenuse is always the longest side and always opposite the right angle. If you are given sides of 6, 8, and 10, the hypotenuse is 10 (not 6 or 8). Check: 6² + 8² = 36 + 64 = 100 = 10².
Forgetting to take the square root. After computing a² + b², you get c², not c. The final step is always to take the square root. 3² + 4² = 25, so c = √25 = 5, not 25.
Solve any right triangle instantly. Use the free Pythagorean Theorem Calculator to find missing sides, the Right Triangle Calculator for angles and areas, and the Distance Calculator for coordinate geometry — no signup required.
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