Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem: a² + b² = c².
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Common Pythagorean Triples
The Pythagorean Theorem
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. This fundamental relationship was known to ancient civilizations and is named after the Greek mathematician Pythagoras.
a² + b² = c²
Pythagorean Triples
A Pythagorean triple consists of three positive integers a, b, and c that satisfy the theorem. The smallest and most famous triple is (3, 4, 5).
| a | b | c | Verification |
|---|---|---|---|
| 3 | 4 | 5 | 3² + 4² = 9 + 16 = 25 = 5² |
| 5 | 12 | 13 | 5² + 12² = 25 + 144 = 169 = 13² |
| 8 | 15 | 17 | 8² + 15² = 64 + 225 = 289 = 17² |
| 7 | 24 | 25 | 7² + 24² = 49 + 576 = 625 = 25² |
Real-World Applications
- Construction: Ensuring corners are square (3-4-5 triangle method)
- Navigation: Calculating straight-line distances
- Architecture: Designing roof pitches and stairs
- Computer Graphics: Calculating distances between points
Extended Pythagorean Triples Reference
A Pythagorean triple is any set of three positive integers (a, b, c) that satisfies a² + b² = c². There are infinitely many such triples. Primitive triples have no common factor greater than 1. Any whole-number multiple of a primitive triple is also a valid triple — for example, multiplying (3, 4, 5) by 2 gives (6, 8, 10), which also satisfies 36 + 64 = 100.
| a | b | c | Primitive? | Check: a² + b² = c² |
|---|---|---|---|---|
| 3 | 4 | 5 | Yes | 3² + 4² = 25 = 5² |
| 5 | 12 | 13 | Yes | 5² + 12² = 169 = 13² |
| 6 | 8 | 10 | No | 6² + 8² = 100 = 10² |
| 8 | 15 | 17 | Yes | 8² + 15² = 289 = 17² |
| 7 | 24 | 25 | Yes | 7² + 24² = 625 = 25² |
| 9 | 40 | 41 | Yes | 9² + 40² = 1681 = 41² |
| 20 | 21 | 29 | Yes | 20² + 21² = 841 = 29² |
| 12 | 35 | 37 | Yes | 12² + 35² = 1369 = 37² |
| 11 | 60 | 61 | Yes | 11² + 60² = 3721 = 61² |
| 13 | 84 | 85 | Yes | 13² + 84² = 7225 = 85² |
The Distance Formula — Pythagorean Theorem in Coordinate Geometry
The distance formula is a direct application of the Pythagorean theorem in a coordinate plane. Given two points A(x₁, y₁) and B(x₂, y₂), the horizontal distance is |x₂ − x₁| and the vertical distance is |y₂ − y₁|. These form the two legs of a right triangle, with the straight-line distance AB as the hypotenuse:
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Example: Find the distance between points A(2, 3) and B(7, 15).
d = √((7 − 2)² + (15 − 3)²)
d = √(5² + 12²)
d = √(25 + 144)
d = √169 = 13
This formula is foundational in computer science (pixel distances in graphics), machine learning (k-nearest neighbors, Euclidean distance), GPS systems, and any field dealing with spatial data. In 3D space, the formula extends naturally: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²).
Step-by-Step Examples
Example 1: Finding hypotenuse (construction)
A carpenter needs to cut a diagonal brace for a 6 ft × 8 ft rectangular frame. How long should the brace be?
c = √(a² + b²)
c = √(6² + 8²) = √(36 + 64) = √100 = 10 ft
The brace should be exactly 10 feet — a perfect (6, 8, 10) triple (a multiple of 3-4-5).
Example 2: Finding a missing leg (ladder problem)
A 13-meter ladder leans against a wall. The base of the ladder is 5 meters from the wall. How high up the wall does the ladder reach?
b = √(c² − a²)
b = √(13² − 5²) = √(169 − 25) = √144 = 12 meters
The ladder reaches 12 meters up the wall — the (5, 12, 13) Pythagorean triple.
Example 3: Checking if a triangle is right-angled
A triangle has sides of length 9, 40, and 41. Is it a right triangle?
a² + b² = 9² + 40² = 81 + 1600 = 1681
c² = 41² = 1681
Since 1681 = 1681, YES — it is a right triangle.
History and Proofs of the Pythagorean Theorem
The Pythagorean theorem is one of the most proved theorems in mathematics — over 370 distinct proofs have been documented. While the theorem bears the name of the Greek mathematician Pythagoras of Samos (c. 570–495 BC), archaeological evidence shows it was known to Babylonian mathematicians over 1,000 years earlier (the Plimpton 322 tablet, c. 1800 BC lists numerous Pythagorean triples), and to ancient Indian mathematicians in the Sulba Sutras (c. 800 BC).
The most intuitive proof uses geometric rearrangement: place four identical right triangles inside a large square of side (a + b). The remaining area in the center can be arranged to form either one square of area c², or two squares of areas a² and b², demonstrating that a² + b² = c².
In 1876, US President James Garfield — while still a congressman — discovered an original proof using a trapezoid. This makes the Pythagorean theorem one of the few mathematical results to have a proof by a head of state. Today, the theorem serves as the foundation for Euclidean geometry, trigonometry, vector spaces, and much of modern mathematics and physics.