Distance Formula Calculator
The straight-line distance between two points — the differences squared, summed, and rooted, with every step printed and the third dimension one optional pair away.
Distance Formula Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Straight-line (Euclidean) only — no road, grid or great-circle distances
- Flat space — no latitude/longitude curvature
In short: From (0, 0) to (3, 4) the distance is exactly 5: the walk is 3 across and 4 up, the squares are 9 and 16, and √25 = 5. That is the 3-4-5 triangle wearing coordinates — the same right triangle behind the slope page’s rise and run and the vector page’s magnitude. The formula never cares about direction: (1, 2) to (4, 6) and (4, 6) to (1, 2) are the same 5, because each difference is squared before it is summed.
Formula
d = √((x₂ − x₁)² + (y₂ − y₁)²)
3D: add (z₂ − z₁)² under the same root
Pythagoras in disguise — the differences are the legs, the distance is the hypotenuse.
Worked Example
- Differences. subtract in the same order on both sides: x₂ − x₁ and y₂ − y₁. Flipping the order changes signs, and squaring erases them — the distance is direction-free.
- Squares. each difference is squared, so every contribution is positive and nothing can cancel.
- Root. the square root of the sum. For the default pair the sum is 25 and the root lands on exactly 5.
(0, 0) to (3, 4): Δx = 3, Δy = 4, 9 + 16 = 25, √25 = 5. Decimals work the same way: (0, 0) to (3, 4.5) gives √29.25 ≈ 5.408330. A zero is an answer too — identical points are 0 apart, and the page says so instead of refusing.
Strengths & Limits Of This Model
Where this engine is strong
- Every step printed: differences, squares, sum, root
- 3D costs two optional fields, nothing else
Where it stops
- No path or polyline distance — chain the page for that
- No lat/lon geodesic mode
Practical Use Cases
Map points
grid references to straight-line separation, before any road geometry
Graphics
hit tests and sprite separation on a canvas
Homework
the working is printed step by step, not just the number
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Distance Formula Calculator — 9 Expert FAQs
9 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why does squaring the differences make direction irrelevant?
A walk of −3 and a walk of +3 cover the same ground. Squaring turns both into 9, so the formula measures only how far, never which way. That is also why the point order cannot change the answer: reversing it flips both signs and the squares erase the flip.
How is this Pythagoras?
The segment between the two points is the hypotenuse of a right triangle whose legs run along the axes: one leg is the x-difference, the other the y-difference. The formula is exactly the Pythagorean theorem with the differences in the leg slots — which is why a 3-across-4-up walk lands on a hypotenuse of 5.
When do I need the z coordinates?
When both points live in 3D space. The law extends by one more square under the same root. Leave both z fields empty and the page stays flat 2D; give one without the other and it refuses, because a half-specified dimension would quietly become zero and fake an answer.
Is the distance ever negative?
No — a distance cannot be. The squares are non-negative, so the root is non-negative. The only way to reach 0 is for every difference to be 0, meaning the two points coincide, and the page names that case rather than hiding it.
Can I feed decimal or negative coordinates?
Yes, freely. Negatives are parenthesized inside the squared terms so the working reads cleanly, and decimals carry through: (0, 0) to (3, 4.5) is √29.25, printed as 5.408330. The arithmetic never rounds until the final display.
How is this different from the slope page?
Slope asks how STEEP the segment is — rise over run, a ratio that ignores how long the segment is. Distance asks how LONG it is. A segment of slope 4/3 can have any length; the (0,0)–(3,4) segment has length exactly 5. The two pages share the 3-4-5 triangle but answer different questions.
What about the vector page’s magnitude?
A magnitude IS a distance — from the vector’s tip to the origin. Type the vector (3, 4) on the vector page and its magnitude prints 5, the same number this page prints for (0, 0) to (3, 4). One law, two costumes.
Why print the working instead of just the number?
Because transcription errors live in one step. A mistyped coordinate shows up as the wrong square, not as a mysteriously wrong root — the working points at where the number went wrong instead of leaving the figure unexplained.
How precise is the printed value?
Six decimal places, with exact values kept exact when they land on them (the default prints 5, not 5.000000). Past about 10⁵⁰ the page switches to E-notation rather than print grouped digits that pretend more precision than the arithmetic still holds.