Your result
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Enter your values, then click Calculate result.How this calculator helps
The distance formula calculator finds Euclidean separation between supplied Cartesian coordinates. Enter two points in three dimensions to compare spatial separation with their two-dimensional XY projection. For planar coordinates, set both z entries to zero. This page does not turn latitude and longitude into road distances, choose a travel route or infer coordinate units from a map.
How to use it
- 1
Enter the first point x, y and z in one consistent Cartesian unit.
- 2
Enter the second point in the same frame; use zero z coordinates for a 2D problem.
- 3
Click Calculate result and compare spatial distance with the planar XY projection.
- 4
Keep the axis unit and frame description with any copied length or displacement vector.
Formula and methodology
d = sqrt((x2−x1)^2 + (y2−y1)^2 + (z2−z1)^2). Planar XY distance omits the z difference. For a 2D problem, set both z coordinates to zero.
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
The points (1, 2, 3) and (4, 6, 15) differ by (3, 4, 12). Their spatial distance is sqrt(9 + 16 + 144) = 13 units. The planar XY distance is sqrt(9 + 16) = 5 units. If both z coordinates are zero, the worksheet returns the two-dimensional result 5 for the corresponding XY points. This example distinguishes a plan view from a full spatial measurement.
How to interpret your result
Spatial distance is a magnitude in the same linear units as the coordinate axes. Squared distance is in square units and is not another ordinary length. The signed coordinate differences show direction along each axis, while the magnitude removes that sign. A correct spatial result cannot establish a valid survey, physical clearance or route unless the source frame and measurements are appropriate.
For different inputs or formulas, use Midpoint Calculator; Slope Calculator; Right Triangle Calculator.
Related questions this calculator covers
- distance formula calculator
- distance between two points calculator
- distance between two points solver
Scenario comparison
| Scenario | What it shows |
|---|---|
| Same point | identical coordinates produce zero distance. |
| Vertical separation | equal x and y but differing z gives zero planar distance. |
| Reversal | swapping the complete points preserves distances and reverses signed differences. |
Common mistakes to avoid
- Mixing axis units or coordinate frames.
- Using latitude and longitude as Cartesian lengths.
- Reading straight-line separation as the length of a traveled route.
Use the 3-4-12 example and independently square each difference to obtain 169 and its square root 13. Verify the XY projection using the 3-4-5 triangle. Swap both complete points and confirm unchanged magnitudes. Translate both points by the same small offset and check that the result is preserved.
Authoritative reference. Method reference checked for this worksheet. The calculation and examples are independently implemented; read the specific scope and units above.What can affect the result?
One coordinate system
Both points must be expressed in the same Cartesian reference frame with consistent units on every axis. A difference of three in x can be combined with a difference of four in y only when the axis scales represent the same distance unit. Converting one axis to metres while leaving another in feet produces a meaningless geometric length. Origin choice may differ between diagrams, so check the reference before copying coordinates.
2D and 3D outputs
The spatial distance uses x, y and z differences. The planar XY output is the projection onto the x-y plane and is no larger than spatial distance. To solve a purely two-dimensional problem, set z1 and z2 to zero. An equal nonzero z coordinate also cancels. A height difference changes spatial distance while leaving the plan projection unchanged. This distinction matters when a diagram contains an elevation difference.
Signed coordinates
Coordinates can be positive, negative or zero. Negative coordinates identify positions relative to the origin; they do not make a distance negative. Each difference is squared before being added, so reversing the complete point order preserves both distances. Changing the order for only one component may preserve a square accidentally but corrupt the displayed displacement vector. Keep each point intact when checking a calculation.
Displacement versus distance
The three signed differences form the displacement from the first point to the second. Its magnitude is the straight-line spatial distance. The actual length of a path can be greater when a route bends, climbs along terrain or follows corridors. This worksheet does not integrate a path or apply a network graph. A coordinate distance can be a lower-bound comparison for a route, but is not a travel-time prediction.
Geographic and measurement limits
Longitude degrees and latitude degrees are angular coordinates on an Earth model, not equal Cartesian metres. Use a geodesic method for geographic separation. The worksheet also does not apply map projections, survey adjustments or instrument uncertainty. Supplied coordinate values are bounded finite numbers. Floating-point rounding remains possible when large absolute coordinates differ by a very small quantity. Local reference coordinates can reduce cancellation when the source data support them.
Geometry checks
For coincident points, distance is zero. If only x changes, distance is the absolute x difference. Translating both points by the same vector leaves their separation unchanged. Rotating a genuine Cartesian frame preserves Euclidean distance, although this interface does not perform the rotation itself. Those invariants provide independent checks that are more useful than comparing two calculators that may share the same implementation or mistaken unit assumptions.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
Use only the disclosed input domains and model. Mixing axis units or coordinate frames. Review the method-specific boundaries above and the linked source before interpreting a result. Calculations use browser floating-point arithmetic; displayed digits are not a measure of real-world certainty. Your entries stay on this device and are not submitted to a calculation server.
Sources and review information
Frequently asked questions
How do I solve a two-dimensional problem?＋
Set both z coordinates to zero. Spatial and XY distances then agree.
Can coordinates be negative?＋
Yes. Signed coordinates are valid; Euclidean distance remains nonnegative.
Does this calculate driving distance?＋
No. It returns a straight-line Cartesian distance and does not use roads or routing data.
Can I paste latitude and longitude?＋
No. Angular geographic coordinates require a geographic distance method, not this Cartesian formula.
Why are the two distances different?＋
A z difference contributes to spatial distance but not to the XY projection.