Solve any right triangle using the legendary a² + b² = c². Compute the hypotenuse or either leg, visualize the triangle, and explore detailed step‑by‑step solutions. Includes historical context, multiple proofs, and real‑world applications.
The Pythagorean theorem is one of the most fundamental and celebrated results in all of mathematics. It states that in a right triangle — a triangle containing a 90° angle — the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides (the legs). In algebraic form:
a² + b² = c²
where c is the hypotenuse, and a and b are the legs.
Named after the ancient Greek mathematician Pythagoras of Samos (c. 570 – c. 495 BC), this theorem has been known and used for millennia. However, evidence suggests that Babylonian and Indian mathematicians were aware of the relationship centuries earlier. The theorem is a cornerstone of Euclidean geometry and has countless applications in science, engineering, architecture, and everyday life.
For any right triangle, the area of the square built on the hypotenuse is equal to the sum of the areas of the squares built on the other two sides. This geometric interpretation is often visualized with squares attached to each side of the triangle — a classic illustration that makes the theorem intuitively clear.
While the theorem bears Pythagoras's name, historical research reveals that the relationship was known to the ancient Babylonians as early as 1800 BC. The Plimpton 322 clay tablet contains a list of Pythagorean triples — integer solutions to a² + b² = c² — demonstrating that Babylonian scribes had a working knowledge of the theorem long before Pythagoras. Similarly, the Shulba Sutras (Indian texts from around 800–600 BC) describe the theorem in the context of altar construction.
Pythagoras and his followers are credited with the first formal proof of the theorem, though no written record survives. The earliest known proof is found in Euclid's Elements (c. 300 BC), where it appears as Proposition 47 of Book I. Since then, hundreds of distinct proofs have been discovered — including one by U.S. President James A. Garfield — making it one of the most-proved theorems in history.
Given: a right triangle with legs a and b, and hypotenuse c.
Theorem: a² + b² = c²
To find the hypotenuse: c = √(a² + b²)
To find a leg: a = √(c² − b²) or b = √(c² − a²)
The theorem can be proven in numerous ways. One of the most elegant is the area rearrangement proof: arrange four identical right triangles inside a square to show that the area of the large square equals the sum of the areas of the smaller squares. This visual proof is often used in classrooms because it requires no algebra — just geometry and spatial reasoning.
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². These triples have fascinated mathematicians for centuries. The smallest and most famous triple is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Euclid provided a formula for generating all primitive triples: for integers m > n > 0, with m and n coprime and of opposite parity, set a = m² − n², b = 2mn, c = m² + n².
Pythagorean triples have practical applications in construction, surveying, and computer graphics. They also appear in number theory, where they relate to Fermat's Last Theorem and the study of elliptic curves.
| Triple | a | b | c | Area | Perimeter |
|---|---|---|---|---|---|
| 3‑4‑5 | 3 | 4 | 5 | 6 | 12 |
| 5‑12‑13 | 5 | 12 | 13 | 30 | 30 |
| 8‑15‑17 | 8 | 15 | 17 | 60 | 40 |
| 7‑24‑25 | 7 | 24 | 25 | 84 | 56 |
| 20‑21‑29 | 20 | 21 | 29 | 210 | 70 |
| 9‑40‑41 | 9 | 40 | 41 | 180 | 90 |
Builders and carpenters use the Pythagorean theorem daily to ensure square corners and precise layouts. The "3‑4‑5 rule" is a classic technique: to check that a corner is square, measure 3 units along one wall, 4 units along the other, and verify that the diagonal is exactly 5 units. This simple application prevents crooked foundations and misaligned structures. Our calculator can instantly verify any set of measurements, saving time and reducing errors.
In navigation, the Pythagorean theorem is used to calculate direct distances between two points given their horizontal and vertical displacements. GPS receivers apply the theorem (in three dimensions) to compute your position from satellite signals. Pilots and sailors use it to plan routes and estimate travel times. Our interactive tool helps visualize these distance calculations in a simple, two‑dimensional setting.
Euclid's proof is a masterpiece of geometric reasoning. He constructs squares on each side of the right triangle and then uses congruence and area arguments to show that the square on the hypotenuse equals the sum of the other two squares. The proof is entirely visual and relies on the concept of equal areas of triangles within the squares.
In 1876, James A. Garfield — then a U.S. Congressman, later President — published a proof using a trapezoid. He arranged two identical right triangles and a smaller right triangle to form a trapezoid, then computed the area in two different ways. The result is a concise algebraic proof that is accessible to high school students.
This proof uses four congruent right triangles arranged within a square. By comparing the area of the large square with the sum of the areas of the four triangles and the smaller central square, one obtains a² + b² = c². This is perhaps the most intuitive and visually appealing proof.