Pythagorean Theorem Calculator
Use the Pythagorean theorem (a² + b² = c²) to find the hypotenuse or any side of a right triangle when you know the other two sides.
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Formula
c = √(a² + b²)
The hypotenuse c is the side opposite the right angle. Square both legs (a and b), add them, then take the square root. To find a leg when the hypotenuse is known: a = √(c² − b²). This only works for right triangles (one 90° angle).
How to use the Pythagorean Theorem Calculator
- 1
Enter your side a
Value should be in units.
- 2
Enter your side b
Value should be in units.
- 3
Read your results instantly
Results update in real time as you type.
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History of the Pythagorean theorem
The theorem is named after Pythagoras of Samos (~570–495 BC), though the relationship was known to Babylonian and Indian mathematicians centuries earlier. The Babylonian clay tablet Plimpton 322 (dated ~1800 BC) lists Pythagorean triples — integer solutions to a² + b² = c².
Euclid provided a formal proof in Elements (Book I, Proposition 47) around 300 BC. There are now over 370 distinct proofs of this theorem, including one by US President James Garfield.
Pythagorean triples
A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a² + b² = c². The most famous is (3, 4, 5): 9 + 16 = 25. Other common triples: (5, 12, 13), (8, 15, 17), (7, 24, 25). Any multiple of these is also a triple: (6, 8, 10), (9, 12, 15), etc.
Pythagorean triples are used in construction because they guarantee a perfect right angle — no protractor or angle tool needed.
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Real-world applications
The Pythagorean theorem is used constantly in construction, navigation, engineering, and physics. Carpenters use the 3-4-5 rule to square walls. GPS systems use it (extended to 3D) to calculate distance. Architects use it to design roof pitches and stair stringers.
In computer graphics, distance calculations between pixels or 3D points rely on the Pythagorean theorem extended to two or three dimensions: distance = √(Δx² + Δy² + Δz²).
Tips & Insights
The 3-4-5 rule in construction
To check if a corner is square: measure 3 feet along one wall, 4 feet along the other. If the diagonal is exactly 5 feet, the corner is a perfect 90°. Scale up for larger structures: 6-8-10, 9-12-15, etc.
Distance formula is Pythagoras in disguise
The distance between two points (x₁,y₁) and (x₂,y₂) is √((x₂−x₁)² + (y₂−y₁)²) — exactly the Pythagorean theorem applied to the horizontal and vertical distances between the points.
Worked Examples
Classic 3-4-5 triple
Hypotenuse: exactly 5 units. Area: 6 square units. Perimeter: 12 units. This is the simplest and most well-known Pythagorean triple.
Screen diagonal calculation
Diagonal: 18.36 inches. This is why a 16:9 aspect ratio monitor with a 27" diagonal has a width of ~23.5" and height of ~13.2".
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Frequently Asked Questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
Does the Pythagorean theorem work for all triangles?
No — only for right triangles (triangles with one 90° angle). For other triangles, use the Law of Cosines: c² = a² + b² − 2ab×cos(C).
What are common Pythagorean triples?
The most common: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Any multiple of these also works: (6,8,10), (9,12,15), (10,24,26), etc.
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