Find any missing side of a right triangle — with step-by-step solutions and visual proof
A Pythagorean triple is a set of three positive integers where a² + b² = c² — meaning all three sides are whole numbers with no decimals. These are the building blocks of right-triangle geometry.
| Side a | Side b | Hypotenuse c | Verify |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 ✓ |
| 5 | 12 | 13 | 25 + 144 = 169 ✓ |
| 8 | 15 | 17 | 64 + 225 = 289 ✓ |
| 7 | 24 | 25 | 49 + 576 = 625 ✓ |
| 20 | 21 | 29 | 400 + 441 = 841 ✓ |
| 9 | 40 | 41 | 81 + 1600 = 1681 ✓ |
| 11 | 60 | 61 | 121 + 3600 = 3721 ✓ |
Any integer multiple of a triple is also a triple. For example, doubling (3, 4, 5) gives (6, 8, 10), and tripling it gives (9, 12, 15). This means infinitely many Pythagorean triples exist.
Builders use the 3-4-5 rule to verify corners are perfectly square. Measure 3 feet along one wall and 4 feet along the other — if the diagonal is exactly 5 feet, the corner is a true 90° angle. This technique has been used for thousands of years.
GPS systems calculate the straight-line distance between two points using the Pythagorean theorem. If you walk 300 meters east and 400 meters north, the direct distance back to your starting point is 500 meters.
OSHA recommends a 4:1 ratio for ladder placement. A 20-foot ladder should have its base 5 feet from the wall. The theorem confirms the ladder reaches √(20² − 5²) = √375 ≈ 19.4 feet up the wall.
TV and monitor sizes are measured diagonally. A display that's 48 inches wide and 27 inches tall has a diagonal of √(48² + 27²) = √3033 ≈ 55 inches. That's how manufacturers get the advertised screen size.
The square on the hypotenuse (green, area 25) equals the sum of the squares on the two legs (blue area 9 + red area 16 = 25).
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Written as a² + b² = c², it's one of the most fundamental relationships in geometry. Discovered by the ancient Greek mathematician Pythagoras around 500 BCE, though Babylonian mathematicians knew the relationship over 1,000 years earlier.
To find the hypotenuse: (1) Square side a, (2) Square side b, (3) Add the results, (4) Take the square root. For example, if a = 3 and b = 4: 3² + 4² = 9 + 16 = 25, and √25 = 5. So the hypotenuse is 5. This calculator handles all the math and shows each step.
No — it only applies to right triangles (triangles with one 90° angle). For other triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C). When angle C is exactly 90°, cos(90°) = 0, and the formula simplifies back to a² + b² = c². So the Pythagorean theorem is actually a special case of the Law of Cosines.
A Pythagorean triple is a set of three positive integers (a, b, c) where a² + b² = c². The smallest and most famous is (3, 4, 5). Others include (5, 12, 13) and (8, 15, 17). Any integer multiple of a triple is also a triple — so (6, 8, 10) and (9, 12, 15) also qualify. There are infinitely many primitive triples (where a, b, c share no common factor).
While named after the Greek mathematician Pythagoras (c. 570–495 BCE), the relationship was known much earlier. The Babylonian clay tablet Plimpton 322 (c. 1800 BCE) contains Pythagorean triples, and ancient Indian mathematicians documented the theorem in the Sulba Sutras (c. 800 BCE). Chinese mathematicians proved it independently in the Zhoubi Suanjing. Pythagoras (or his school) is credited with the first known formal proof.