Worked example
Three-four-five triangle
- Input
- a = 3; b = 4
- Output
- c = 5
The squares sum to 9 + 16 = 25.
Pythagorean Theorem Calculator
Choose the side to solve and enter the other two positive side lengths. The calculator validates that an entered hypotenuse exceeds the known leg before applying the Pythagorean theorem. This page covers “pythagoras equation solver”, “hypotenuse calculator”.
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How to use this calculator
Enter the side to solve and the other two positive side lengths in the same unit.
Review which side is the hypotenuse, consistent units, measurement precision, and whether the triangle is known to contain a 90° angle.
Read both legs, the hypotenuse, and triangle area.
Formula
For legs a and b, c = √(a² + b²). When c and one leg are known, the other leg is √(c² − known-leg²). Area = a × b ÷ 2.
Worked example
A right triangle with legs 3 and 4 has hypotenuse √(3² + 4²) = 5.
Reference guide
Formula details and result guidance
Follow the variables, derivation, precision rule, and worked cases below to reproduce the calculation independently.
Variables
| Symbol | Meaning | Unit | Role |
|---|---|---|---|
| a, b | Perpendicular side lengths | selected length unit | Input |
| c | Hypotenuse opposite the right angle | selected length unit | Output |
Derivation
- For a right triangle, the squared side relationship is a² + b² = c².
- Solve the hypotenuse with c = √(a² + b²), or a perpendicular side with a = √(c² − b²).
- A perpendicular-side solution requires a positive hypotenuse that is longer than the known perpendicular side.
Rounding and precision
The square-root calculation uses unrounded inputs and shows up to 6 decimal places. Keep more precision for chained layout work and round only at the final measurement step.
Worked and boundary examples
Worked example
Missing perpendicular side
- Input
- c = 13; b = 5
- Output
- a = 12
Subtract 25 from 169, then take √144.
Boundary case
Nearly flat valid triangle
- Input
- c = 5; b = 4.999
- Output
- a ≈ 0.099995
The radicand is positive but small; measurement error becomes important here.
Common mistakes
- Applying the theorem to a triangle that is not known to contain a right angle.
- Adding unsquared sides or forgetting the square root after summing squares.
- Entering a known perpendicular side that equals or exceeds the hypotenuse when solving for the other perpendicular side.
How to interpret the result
The result is a geometric side length for an ideal right triangle. It does not prove that field corners are square or account for material thickness and measurement tolerances.
Scope
Appropriate for
- Checking a right-triangle diagonal from perpendicular runs.
- Finding one missing side when the other two valid sides are known.
Do not use for
- Solving an oblique triangle without a right angle.
- Certifying a structural layout or survey from nominal dimensions alone.
Tools for the same task
Next tools in the workflow
Evidence and sources
- In a right triangle, the sum of the squared perpendicular sides equals the squared hypotenuse.OpenStax: College Algebra 2e — Preface and ScopeSupports: Slope, equations, radicals, proportions, and inverse algebraic operations. Reviewed 2026-08-22.
- Solving a perpendicular side requires a nonnegative difference of squares.OpenStax: Prealgebra 2e — Preface and ScopeSupports: Percent, proportion, radical, ratio, and elementary geometry procedures. Reviewed 2026-08-22.
Assumptions and limits
- The theorem applies only to a Euclidean right triangle; measured construction corners may require a tolerance rather than exact equality.
- Pythagorean Theorem Calculator treats the entered values as estimates and does not infer missing project, account, or personal details.
- Calculations retain working precision except where the cited method requires an intermediate rounding step; displayed values are then formatted separately. Source rules and dated rates must be rechecked when they change.
Sources
- OpenStax: College Algebra 2eAlgebraic, ratio, radical, equation, and function formulas. Reviewed 2026-08-14.
- OpenStax: Elementary Algebra 2eOfficial formula or method source used by this calculator. Reviewed 2026-08-14.
Related calculators
Verification
How this calculator is checked
Trust comes from reproducible evidence, not model confidence. You can inspect the formula, assumptions, worked example, and cited sources on this page.
- Versioned calculationFormula logic is kept separate from the interface and covered by automated registry and behavior checks.
- Cited evidenceSources linked for independent checking: 2.
- Transparent AI useAI may assist drafting or adversarial review. Agreement between models is not proof, and no human expert review is claimed unless a named reviewer is shown.
Frequently asked questions
How does the Pythagorean Theorem Calculator work?
For legs a and b, c = √(a² + b²). When c and one leg are known, the other leg is √(c² − known-leg²). Area = a × b ÷ 2.
What information should I enter in the Pythagorean Theorem Calculator?
Enter the side to solve and the other two positive side lengths in the same unit, then verify which side is the hypotenuse, consistent units, measurement precision, and whether the triangle is known to contain a 90° angle.
What should I verify before using this Pythagorean Theorem Calculator result?
The theorem applies only to a Euclidean right triangle; measured construction corners may require a tolerance rather than exact equality.
Last updated 2026-08-25 · any-two-sides-v2 · content-2026-08-14 · Published by YunFanLabs