First point P1 = (1, 2, 3). Second point P2 = (4, 6, 15).
MATH / ANALYTIC GEOMETRY
3D Distance Calculator
Calculate Euclidean distance between two three-dimensional points. Inspect the equation, substituted values, result, and interpretation limits in one guided workspace.
- 01 Calculated in this tab
- 02 Values stay in this browser tab
- 03 Use boundary
AXIS OFFSETS / BOX DIAGONAL
Check the straight-line gap, not just the final number
This explainer is built for the part of 3D distance problems that is easiest to skip: the coordinate differences. With the reviewed default points (1, 2, 3) and (4, 6, 15), the calculator first isolates dx = 3, dy = 4, and dz = 12, then squares and adds those offsets to get 9 + 16 + 144 = 169 before taking the square root. That is why the page returns a straight-line distance of 13.
Worked default example
- Entered points
- First point P1 = (1, 2, 3). Second point P2 = (4, 6, 15).
- Axis-by-axis changes
- dx = x2 - x1 = 4 - 1 = 3. dy = y2 - y1 = 6 - 2 = 4. dz = z2 - z1 = 15 - 3 = 12.
- Squared offsets
- 3^2 = 9, 4^2 = 16, and 12^2 = 144.
- Sum before the root
- 3^2 + 4^2 + 12^2 = 9 + 16 + 144 = 169.
- Returned 3D distance
- d = sqrt(3^2 + 4^2 + 12^2) = sqrt(169) = 13.
What to enter and how to read the result
Enter x, y, and z for the first point and then x, y, and z for the second point, all on one Cartesian coordinate system and one consistent unit scale. The main result card reports the straight-line segment length between those two points. The formula panel underneath the workspace shows the substituted equation so you can verify each coordinate difference before using the distance elsewhere.
What the calculator actually does
Nirmion evaluates the 3D Euclidean distance formula d = sqrt[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]. Because the formula depends only on coordinate differences, translating the first point to the origin leaves the length unchanged. In the reviewed default example, (1, 2, 3) to (4, 6, 15) becomes O to (3, 4, 12), so d = sqrt(3^2 + 4^2 + 12^2) = sqrt(169) = 13.
Where this model stops helping
Use this page only when both points live in the same Cartesian coordinate system and the same unit scale. It does not calculate travel path length, road distance, GPS geodesic distance on Earth, or distances between points recorded in mixed units or incompatible projections. If your coordinates come from latitude and longitude, convert or project them appropriately before using this model.
Reference and next checks
The linked OpenStax College Algebra 2e source covers Cartesian distance formulas. If you need the midpoint of the same segment, the distance from a point to a plane, or the length of a 3D vector after translation, continue with the related calculators below.
Read OpenStax College Algebra 2eTranslate by subtracting P1
(1, 2, 3) to (4, 6, 15) has the same length as O to (3, 4, 12).
3D distance as a rectangular walk and one straight segment
A stepped path showing x shift 3, y shift 4, and z shift 12 from the translated origin to (3, 4, 12), plus a diagonal segment labeled d = 13.
dx = x2 - x1 = 4 - 1 = 3. dy = y2 - y1 = 6 - 2 = 4. dz = z2 - z1 = 15 - 3 = 12.
3^2 + 4^2 + 12^2 = 9 + 16 + 144 = 169.
d = sqrt(3^2 + 4^2 + 12^2) = sqrt(169) = 13.
Continue the 3D geometry workflow
Continue the 3D geometry workflow
COMMON QUESTIONS
3D distance calculator FAQs
Can I enter negative coordinates on this calculator?
Yes. Negative coordinates are valid as long as both points use the same Cartesian system and unit scale. The formula uses coordinate differences, so the sign matters only through the offset between the two points.
Why does the worked example return exactly 13?
The default points produce dx = 3, dy = 4, and dz = 12. Their squares add to 169, so the square root is exact: d = sqrt(3^2 + 4^2 + 12^2) = sqrt(169) = 13.
Does this page work for latitude and longitude or driving distance?
No. This is a Cartesian straight-line calculator. Latitude and longitude need a geodesic or projected-distance workflow, and travel routes need path-based mapping data rather than the Euclidean formula.
Use boundary
Calculation path
Calculate Euclidean distance between two three-dimensional points. The page evaluates the displayed equation from your supplied values and presents both symbolic and substituted KaTeX working so the arithmetic can be checked.
Calculation path
d=sqrt[(x2-x1)^2+(y2-y1)^2+(z2-z1)^2]
What you provide
What you provide
- First point x, using the same scenario as the other inputs
- First point y, using the same scenario as the other inputs
- First point z, using the same scenario as the other inputs
- Second point x, using the same scenario as the other inputs
- Second point y, using the same scenario as the other inputs
- Second point z, using the same scenario as the other inputs
What you receive
What you receive
- 3D distance from the stated equation
- A symbolic formula plus substituted working with your values
- Visible assumptions, field guidance, and an authoritative learning reference
Use boundary
Choose the maximum decimal places shown. This does not increase source accuracy.
Coordinates and lengths use one consistent Cartesian scale; rounding is applied only to the displayed result.
Use internally consistent units and retain extra precision when carrying the result into another calculation.
Reviewed reference factors