The Euclidean Distance Calculator determines the straight-line distance between two points in space, whether in two, three, or higher dimensions. This tool is widely used in geometry, machine learning, data science, image processing, and spatial analytics. It helps users quickly find the shortest distance between any two points in Cartesian coordinates.
Whether you’re analyzing spatial relationships, computing distances in a dataset, or performing vector operations, this calculator makes the math easy and precise. It’s especially helpful for anyone working with coordinates, such as GIS analysts, researchers, developers, and engineers.
Formula of Euclidean Distance Calculator
The general formula for Euclidean distance in n-dimensional space is:
d = √[(x₁ − y₁)² + (x₂ − y₂)² + (x₃ − y₃)² + … + (xₙ − yₙ)²]
Where:
- d = Euclidean distance
- x₁, x₂, …, xₙ = coordinates of the first point
- y₁, y₂, …, yₙ = coordinates of the second point
- n = number of dimensions
For 2D space:
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
For 3D space:
d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
This formula calculates the direct, or “as-the-crow-flies,” distance between two points. It assumes all coordinates are in a uniform metric space.
Helpful Reference Table
Here’s a quick reference for commonly calculated 2D and 3D Euclidean distances between point pairs:
| Point A | Point B | Distance (2D) or (3D) |
|---|---|---|
| (0, 0) | (3, 4) | 5.00 |
| (1, 2, 3) | (4, 6, 8) | 7.07 |
| (5, 5) | (5, 5) | 0.00 |
| (2, 3, 1) | (6, 7, 2) | 6.48 |
| (−1, −1) | (1, 1) | 2.83 |
These distances are rounded for convenience and demonstrate how the calculator applies to various point sets.
Example of Euclidean Distance Calculator
Imagine you want to calculate the distance between two points in 3D space:
- Point A = (2, 3, 5)
- Point B = (7, 1, 9)
Step 1: Apply the 3D formula
d = √[(7 − 2)² + (1 − 3)² + (9 − 5)²]
d = √[25 + 4 + 16] = √45 ≈ 6.71
Result: The Euclidean distance is approximately 6.71 units.
This result can be used in engineering measurements, data clustering, or simulation modeling.
Most Common FAQs
Euclidean distance is used to measure the straight-line distance between two points in a coordinate space. It’s a core metric in geometry, clustering algorithms, classification models, and spatial analysis.
It depends. In flat coordinate systems, yes. But in geographic applications, Euclidean distance doesn’t account for Earth’s curvature. In those cases, haversine or geodesic distance formulas are more accurate.
Yes. The Euclidean distance formula works for any number of dimensions. Just apply the same formula using all the coordinates of each point.