A calculator for finding the distance between two points should identify the plane and show the coordinate differences before returning a number. For ordinary Euclidean distance, both axes must describe the same length unit with a common scale. The original fixtures below make the result, symmetry, and zero case easy to check.
Start with a compatible coordinate basis
Enter A = (x₁,y₁) and B = (x₂,y₂) in a two-dimensional rectangular plane. If both axes measure centimeters, the distance will be in centimeters. If the coordinates are abstract but equally scaled, label the result in coordinate units.
An x-axis measured in years and a y-axis measured in dollars does not fit this physical-distance interpretation. Neither do latitude/longitude coordinates merely because both are numbers. This helper does not calculate geodesic distance, map-projection effects, driving routes, or survey measurements.
OpenStax derives the planar distance formula from the Pythagorean theorem. The horizontal and vertical differences form perpendicular legs; their squared lengths add to the square of the straight-line distance. This example uses its own points and proposed diagram.
Show all four calculation steps
Delta x = x₂ − x₁
Delta y = y₂ − y₁
S = (Delta x)² + (Delta y)²
Distance = sqrt(S)
For A = (−2,1) and B = (1,5), subtract carefully: Δx = 1 − (−2) = 3 and Δy = 5 − 1 = 4. The squared sum is 3² + 4² = 9 + 16 = 25. Its nonnegative square root is five.
| Record part | Value |
|---|---|
| A | (−2,1) |
| B | (1,5) |
| Δx / Δy | 3 / 4 |
| Squared sum S | 25 squared coordinate units |
| Straight-line distance | 5 coordinate units |
The signed differences say which way B lies relative to A. Distance is nonnegative because it comes from the sum of squares. The sum can be zero; a squared value is not necessarily positive.
The separate sum |Δx| + |Δy| equals seven here. That is a Manhattan-style distance for horizontal-plus-vertical travel, not the straight-line distance. A real route may follow still another path; the calculator does not infer it.
Swap the points and test special cases
Swapping A and B changes Δx to −3 and Δy to −4. Their squares remain nine and 16, so distance remains five. A useful Swap Points control moves the entire ordered pairs and recomputes the derivation, rather than swapping only the x coordinates.
| A | B | Expected Euclidean result |
|---|---|---|
| (−2,1) | (1,5) | 5 units |
| (1,5) | (−2,1) | 5 units, signed differences reversed |
| (2,−1) | (2,4) | 5 units; Δx = 0 |
| (2,4) | (2,4) | 0 units |
| (0,0) | (1,1) | √2 units, approximately 1.414214 |
Identical points give a legitimate zero-length segment, not an input error. In the diagonal unit-square case, √2 is the exact mathematical expression; the six-place decimal is approximate. The small numeric helper does not promise a general exact radical simplifier.
Keep a fitted diagram honest
An optional original diagram can plot the two endpoints, their connecting segment, and the horizontal and vertical legs. For the first fixture, the corner at (1,1) completes the right triangle, with legs of three and four units.
Use the same displayed length per coordinate unit on both axes. Fit the view by changing its bounds or uniform display scale, never by changing the supplied coordinates or assigning a different unit to one axis. Keep labels readable, include the unit, and show coincident points without inventing a visible nonzero segment.
The diagram illustrates the record; its pixel length is not a measurement of the input distance. Text coordinates and the full derivation remain available even if the graphic cannot be read.
Validate each coordinate before drawing
Accept signed finite dot-decimal coordinates with a leading digit, no more than six fractional places, and magnitude no greater than 10,000. Zero and negative coordinates are valid. A blank coordinate, comma-formatted number, scientific notation, nonfinite value, or out-of-range input is invalid.
Require an explicit acknowledgment that both axes use a common unit and scale; if it is missing, leave the distance unresolved. Do not silently normalize incompatible axes. Any input or unit-context edit clears stale derivation, result, and optional diagram until Calculate succeeds.
Reset empties all coordinate entries, the context acknowledgment, result, and diagram. Keep internal differences and squared sums unrounded, then format the numeric distance to six decimal places with approximation marking. The original coordinates, common unit, and calculation trace belong in the plain-text record.
Copyable distance prompt
Create an original self-contained HTML/CSS/JavaScript planar Euclidean distance helper with readable source and an acceptance-test report. Four native labeled fields x1,y1,x2,y2, common-unit context and explicit same-axis-scale acknowledgment. No latitude/longitude, route, survey, projection or 3D calculations.
Compute dx=x2-x1,dy=y2-y1,S=dx^2+dy^2,d=sqrt(S). Show input ordered pairs, signed differences, squared terms/S with squared unit, and nonnegative distance with common unit. Keep internal values unrounded; display numeric distance to six decimals with approximation marker as needed, no general exact-radical simplifier.
Fixtures: A(-2,1),B(1,5) ->dx 3,dy 4,S 25,d 5. Swap entire points ->dx -3,dy -4,d 5. Vertical(2,-1),(2,4)->5; identical(2,4),(2,4)->0. Unit diagonal(0,0),(1,1)->sqrt2≈1.414214, not an exact decimal. Manhattan 3+4=7 is a different distance.
Inputs signed finite dot decimals with leading digit, up to six fractional places and magnitude<=10000. Zero/negative valid; blank,comma,scientific notation,nonfinite,out-of-range or unconfirmed common basis invalid. Edits clear all stale output/diagram. Native Calculate validates; Swap Points requires a valid record, swaps both pairs and recomputes; Reset blanks entries/context/results.
Optional original equal-unit-axis diagram uses labels, endpoint segment and delta legs, with first-fixture corner(1,1). Fit view uniformly without changing coordinates/units; handle coincident points and preserve complete text fallback. Diagram pixels are not measurement proof.
Test fixtures, symmetry, units, invalid-after-valid clearing, whole-pair Swap, focus, actual article width and 360 px. Mark unexecuted runtime/diagram checks not_tested. No saved progress, export, network, survey accuracy or tested-host promise.
Keep the coordinate record beside the answer
Retain the native inputs and difference trace with any helper source. For an optional prepared HTML version in NOS, its supplied product evidence supports HTML saving and rendering; verify fields, Swap Points, and error clearing in the actual workspace before treating that version as a tested calculator. Explore NOS with prepared HTML, keeping the common unit and squared differences beside the distance so a copied answer still identifies which question was solved.