📐 Pythagorean Theorem Calculator

Find any missing side of a right triangle using a² + b² = c². Choose what you want to find, enter the two known sides, and get the answer with full step-by-step working.

Enter sides a and b to find hypotenuse c.

Side a

Side b

What is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental relationship in Euclidean geometry stating that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Expressed as a² + b² = c², where c is the hypotenuse and a and b are the two legs, this theorem is one of the most widely known and applied results in all of mathematics — used continuously in construction, navigation, engineering, physics, and computer graphics.

The theorem is named after the ancient Greek philosopher and mathematician Pythagoras of Samos (c. 570–495 BCE), though evidence suggests the relationship was known to Babylonian mathematicians more than 1,000 years earlier. Babylonian clay tablets dated to around 1800 BCE contain Pythagorean triples — integer solutions to a² + b² = c² such as (3, 4, 5) and (5, 12, 13) — demonstrating practical knowledge of right-triangle geometry long before Pythagoras is thought to have formally proved it.

The Pythagorean Theorem extends far beyond simple right-triangle calculations. In coordinate geometry, it underpins the distance formula between two points. In trigonometry, it establishes the fundamental identity sin²θ + cos²θ = 1. In three dimensions, the distance formula becomes √(a² + b² + c²). In relativity theory, the spacetime interval uses a modified Pythagorean relationship. The theorem's reach extends to inner product spaces, signal processing (Parseval's theorem), and quantum mechanics — making it arguably the most consequential theorem in the history of mathematics.

Pythagorean Theorem

Hypotenuse: c = √(a² + b²)

Side a: a = √(c² − b²)

Side b: b = √(c² − a²)

Worked Example — Find c with a=3, b=4

a² + b² = c²

3² + 4² = c²

9 + 16 = c²

c² = 25

c = √25 = 5

Area = ½ × 3 × 4 = 6

Perimeter = 3 + 4 + 5 = 12

How the Pythagorean Theorem Calculator Works

Formula, assumptions, and calculation steps for this math tool.

Methodology

Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.

Calculation Steps

Read the values and operation selected.
Normalize signs, decimals, fractions, or units if needed.
Apply the mathematical rule or formula.
Format the answer and any intermediate values for checking.

Assumptions and Limits

Inputs must be within the supported domain of the operation.
Decimal answers may be rounded for readability.
Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

What is 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: a² + b² = c². It was named after the ancient Greek mathematician Pythagoras.

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². The most famous is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple — e.g. (6, 8, 10).

What are real-world uses of the Pythagorean theorem?

It is used in construction (checking square corners), navigation (finding shortest distances), architecture, computer graphics (calculating pixel distances), and physics (resolving vector components). Carpenters use the 3-4-5 rule to ensure right angles.

How do I find the hypotenuse?

Square both known sides, add them together, then take the square root. For example, a=6 and b=8: c = √(36+64) = √100 = 10.

Real-World Applications

🏗️

Construction & Squaring Foundations

Builders use the 3-4-5 rule — a classic Pythagorean triple — to square corners: measuring 3 feet along one wall, 4 feet along the adjacent wall, and confirming the diagonal is exactly 5 feet ensures a perfect 90° corner for foundations, framing, and tiling.

📐

Staircase & Ramp Design

Architects calculate the diagonal stringer length of a staircase from the rise (vertical height) and run (horizontal depth) using a² + b² = c² — determining the total length of timber or steel needed for each stringer and the overall slope angle.

🗺️

Distance Calculation (GPS & Mapping)

Calculate the straight-line distance between two points on a flat map: if two locations are 3 km east and 4 km north of a reference point, the straight-line distance is √(3² + 4²) = 5 km. GPS navigation systems use extended 3D versions for precise distance calculation.

🖥️

Screen Size Calculation

Monitor and TV screen sizes are measured diagonally — a display advertised as "27 inches" has a diagonal of 27 inches. Given a 16:9 aspect ratio, the Pythagorean Theorem calculates the actual width and height: width = 27 × 16/√(16² + 9²) ≈ 23.5 inches.

🔌

Electrical Engineering — Impedance

In AC circuits, total impedance Z = √(R² + X²) where R is resistance and X is reactance — a direct application of the Pythagorean Theorem in the complex impedance plane. This relationship determines current, power factor, and component ratings in electrical design.

🎮

Game Development & Collision Detection

Video game engines calculate the distance between two objects using the 2D or 3D distance formula — derived from the Pythagorean Theorem — to determine whether objects are close enough to trigger collision events, proximity alerts, or targeting acquisition.

Common Mistakes

Applying the theorem to non-right triangles

The Pythagorean Theorem only applies to right-angled triangles — those containing exactly one 90° angle. For any other triangle, the correct tool is the Law of Cosines: c² = a² + b² − 2ab·cos(C). Using a² + b² = c² on a non-right triangle produces an incorrect hypotenuse length.

Incorrectly identifying the hypotenuse

The hypotenuse is always the side opposite the right angle — and it is always the longest side. A common mistake is applying the formula with a leg substituted for c rather than the hypotenuse. Always confirm which side is opposite the right angle before assigning variables.

Forgetting to take the square root

The theorem states a² + b² = c², so c = √(a² + b²). Students frequently compute a² + b² and report that as the length — forgetting the final square root step. The result without the square root is c², not c.

Using the theorem in 3D without the correct extension

The 3D distance between points (x₁,y₁,z₁) and (x₂,y₂,z₂) is √((Δx)² + (Δy)² + (Δz)²) — an extension of the theorem to three dimensions. Using only two dimensions in a 3D problem systematically underestimates the actual distance.

Confusing Pythagorean triples with all right triangles

Pythagorean triples (3-4-5, 5-12-13, 8-15-17) are integer solutions to a² + b² = c². Not all right triangles have integer-side lengths — a right triangle with legs of 1 and 1 has a hypotenuse of √2 (irrational). Pythagorean triples are convenient special cases, not a general requirement.

Common Pythagorean Triples Quick Reference

Triple (a, b, c)	Check (a² + b²)	Common Use
3, 4, 5	9 + 16 = 25	Construction squaring
5, 12, 13	25 + 144 = 169	Surveying, geometry
8, 15, 17	64 + 225 = 289	Engineering problems
7, 24, 25	49 + 576 = 625	Geometry proofs
6, 8, 10	36 + 64 = 100	3-4-5 scaled (×2)
9, 40, 41	81 + 1600 = 1681	Advanced calculations

References

Euclid. Elements, Book I, Proposition 47. c. 300 BCE.
Heath, T.L. The Thirteen Books of Euclid's Elements. Cambridge University Press, 1908.
Loomis, E.S. The Pythagorean Proposition. National Council of Teachers of Mathematics, 1968.
Maor, E. The Pythagorean Theorem: A 4,000-Year History. Princeton University Press, 2007.
Stewart, J. Calculus: Early Transcendentals. Cengage, 2015.