Mathematics · Geometry
3D Distance Calculator
Calculate Euclidean distance between two points in three dimensions.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Change an input to reshape this diagram.
Calculation steps
- Displacement=(2,3,6).
- Distance=√(2²+3²+6²)=7.
Understand 3D distance
One idea, three depths
Choose how deeply to explain 3D distance
3D distance: Calculate Euclidean distance between two points in three dimensions.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using 3D distance to answer this question: calculate euclidean distance between two points in three dimensions? Enter x₁, y₁, z₁, and 3 other inputs; the calculator shows 3D distance. For example: (0,0,0) to (2,3,6) has distance 7. The answer tells you 3D distance.
Age 15Explain it to a 15-year-oldConnect it to the formula
The Pythagorean theorem extends by adding the squared displacement along a third perpendicular axis. The rule is d=√[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]. Its input values are x₁, y₁, z₁, x₂, y₂, z₂, and the main result is 3D distance. For example: (0,0,0) to (2,3,6) has distance 7.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated 3d distance relation over the valid real-number domain stated below. The implemented relation is d=√[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²], evaluated from x₁, y₁, z₁, x₂, y₂, z₂ to produce 3D distance. The Pythagorean theorem extends by adding the squared displacement along a third perpendicular axis. Coordinates must share the same axes and units.
Inputs and valid domain
- x₁ must be a finite real number.
- y₁ must be a finite real number.
- z₁ must be a finite real number.
- x₂ must be a finite real number.
- y₂ must be a finite real number.
- z₂ must be a finite real number.
Important boundary: Coordinates must share the same axes and units.
The formula
d=√[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]
How the calculator works through it
It substitutes x₁, y₁, z₁, x₂, y₂, z₂ into the formula and exposes every numerical step above. The main output is 3D distance, accompanied by Δx, Δy, Δz.
Read the result correctly
The 3D distance is the direct answer to “calculate euclidean distance between two points in three dimensions.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
(0,0,0) to (2,3,6) has distance 7.
Where this model stops being reliable
Coordinates must share the same axes and units.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why 3D distance works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
3D distance uses d=√[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Ratios between measured quantities
Ratios help you check the scale, units and proportional meaning of 3D distance.
Review this foundation about 4 min
Optional enrichment
- Angles and geometric relationships
Angle language provides useful geometric context for extending 3D distance to related shapes and constructions.
Review this foundation about 4 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read x₁, y₁, z₁, x₂, y₂, z₂.
- Evaluate the principal relationship: d=√[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²].
- Return 3D distance and check the domain conditions described above.
Python
from math import *
def distance_3d(x1, y1, v1, x2, y2, v2) -> float:
return sqrt(((((x2 - x1) * (x2 - x1)) + ((y2 - y1) * (y2 - y1))) + ((v2 - v1) * (v2 - v1))))
assert abs(distance_3d(0, 0, 0, 2, 3, 6) - 7) < 1e-6 * max(1.0, abs(7))
C
#include <assert.h>
#include <math.h>
double distance_3d(double x1, double y1, double v1, double x2, double y2, double v2) {
return sqrt(((((x2 - x1) * (x2 - x1)) + ((y2 - y1) * (y2 - y1))) + ((v2 - v1) * (v2 - v1))));
}
int main(void) {
const double expected = 7;
const double actual = distance_3d(0, 0, 0, 2, 3, 6);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double distance_3d(double x1, double y1, double v1, double x2, double y2, double v2) {
return std::sqrt(((((x2 - x1) * (x2 - x1)) + ((y2 - y1) * (y2 - y1))) + ((v2 - v1) * (v2 - v1))));
}
int main() {
constexpr double expected = 7;
const double actual = distance_3d(0, 0, 0, 2, 3, 6);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double distance_3d(double x1, double y1, double v1, double x2, double y2, double v2)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global distance_3d
section .text
distance_3d:
push rbp
mov rbp, rsp
sub rsp, 144
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd [rbp-32], xmm3
movsd [rbp-40], xmm4
movsd [rbp-48], xmm5
movsd xmm0, [rbp-32]
subsd xmm0, [rbp-8]
movsd [rbp-88], xmm0
movsd xmm0, [rbp-32]
subsd xmm0, [rbp-8]
movsd [rbp-96], xmm0
movsd xmm0, [rbp-88]
mulsd xmm0, [rbp-96]
movsd [rbp-80], xmm0
movsd xmm0, [rbp-40]
subsd xmm0, [rbp-16]
movsd [rbp-112], xmm0
movsd xmm0, [rbp-40]
subsd xmm0, [rbp-16]
movsd [rbp-120], xmm0
movsd xmm0, [rbp-112]
mulsd xmm0, [rbp-120]
movsd [rbp-104], xmm0
movsd xmm0, [rbp-80]
addsd xmm0, [rbp-104]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-48]
subsd xmm0, [rbp-24]
movsd [rbp-136], xmm0
movsd xmm0, [rbp-48]
subsd xmm0, [rbp-24]
movsd [rbp-144], xmm0
movsd xmm0, [rbp-136]
mulsd xmm0, [rbp-144]
movsd [rbp-128], xmm0
movsd xmm0, [rbp-72]
addsd xmm0, [rbp-128]
movsd [rbp-64], xmm0
sqrtsd xmm0, [rbp-64]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-56]
leave
ret
MATLAB
function result = distance_3d(x1, y1, v1, x2, y2, v2)
result = sqrt(((((x2 - x1) * (x2 - x1)) + ((y2 - y1) * (y2 - y1))) + ((v2 - v1) * (v2 - v1))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[x1_, y1_, v1_, x2_, y2_, v2_] := Sqrt[((((x2 - x1) * (x2 - x1)) + ((y2 - y1) * (y2 - y1))) + ((v2 - v1) * (v2 - v1)))];
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). 3D Distance Calculator. MW SysArc Tools. https://math.mwsysarc.com/geometry/3d-distance
MLA 9
MW SysArc. “3D Distance Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/geometry/3d-distance. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “3D Distance Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/geometry/3d-distance.
Harvard
MW SysArc (2026) ‘3D Distance Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/geometry/3d-distance (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_distance_3d_2026,
author = {{MW SysArc}},
title = {3D Distance Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/geometry/3d-distance},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - 3D Distance Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/geometry/3d-distance
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the 3D distance do?
Calculate Euclidean distance between two points in three dimensions.
How does the 3D distance work?
The calculator applies d=√[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]. The Pythagorean theorem extends by adding the squared displacement along a third perpendicular axis.
What can I learn from the 3D distance?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .