d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}Variables
- (x₁, y₁): first point
- (x₂, y₂): second point
- d: distance between the points
How to use this formula
Calculates the straight-line distance between two points in a coordinate plane.
Important notes
- The formula follows from the Pythagorean theorem.
- The order of the two points does not change the result.
Quick example
Points (1, 2) and (4, 6) are 5 units apart.
Applicability, worked calculation, and verification
Domain and applicability
Applies to two points in a Euclidean Cartesian plane; the result is always nonnegative and independent of point order.
Units
- both coordinate axes must use compatible length units
Assumptions and domain checks
- The order of the two points does not change the result.
- For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.
- For the Distance Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Distance Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Distance Formula, check zero, negative, and extreme input values before relying on the result.
- When using Distance Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
The distance between (1, 2) and (4, 6) is √[(4 − 1)² + (6 − 2)²] = √(9 + 16) = 5.
- Find the horizontal change: Δx = 4 − 1 = 3.
- Find the vertical change: Δy = 6 − 2 = 4.
- Square and add the changes: 3² + 4² = 25.
- Take the nonnegative square root to obtain a distance of 5.
Independent verification
The coordinate differences form a 3–4–5 right triangle, so the computed straight-line distance of 5 is consistent with the Pythagorean theorem.
Common mistakes
- Do not drop the radical boundary in Distance Formula; verify exactly which terms are inside the root and whether the chosen root is valid.
- For the Distance Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Distance Formula 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
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^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 Distance Formula.
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 Distance Formula used for?
Calculates the straight-line distance between two points in a coordinate plane.
Can I copy this formula as LaTeX?
Yes. Copy d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^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.