a^2+b^2=c^2Variables
- a, b: perpendicular leg lengths
- c: hypotenuse length
How to use this formula
Relates the side lengths of a right triangle.
Important notes
- The theorem applies only to right triangles.
- The hypotenuse c is opposite the right angle and is the longest side.
Quick example
A triangle with legs 3 and 4 has hypotenuse 5.
Applicability, worked calculation, and verification
Domain and applicability
Applies only to Euclidean right triangles; c denotes the side opposite the right angle and must be the longest side.
Units
- a, b, and c must use the same length unit
Assumptions and domain checks
- The theorem applies only to right triangles.
- Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula.
Do not use this formula when
- Do not use Pythagorean Theorem when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units.
Boundary and special cases
- For Pythagorean Theorem, check zero, negative, and extreme input values before relying on the result.
- When using Pythagorean Theorem, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form a^2+b^2=c^2 for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For a right triangle with legs 3 and 4, calculate the hypotenuse: c = √(3² + 4²) = √25 = 5.
- Confirm that the triangle is right-angled and that a and b are the perpendicular legs.
- Square the leg lengths: 3² = 9 and 4² = 16.
- Add the squared lengths: 9 + 16 = 25.
- Take the nonnegative square root to obtain c = 5.
Independent verification
The result forms the known 3–4–5 right triangle, and 5² = 25 equals 3² + 4² = 25.
Common mistakes
- Before substituting values into Pythagorean Theorem, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- For the Pythagorean Theorem, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step.
Continue the workflow
Use Pythagorean Theorem in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
a^2+b^2=c^2. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Pythagorean Theorem.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Passed the core formula indexing gate.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Pythagorean Theorem used for?
Relates the side lengths of a right triangle.
Can I copy this formula as LaTeX?
Yes. Copy a^2+b^2=c^2 or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.