Quick answer
Euclidean distance between points a and b is d = √Σ(aᵢ − bᵢ)². Between (1, 2, 3) and (4, 6, 3): √(9 + 16 + 0) = 5.
Enter two points with the same number of coordinates. The calculator returns the distance.
Related tools: Distance Between Two Points Calculator: Straight-line distance and midpoint between two points in the plane. Area Calculator: Area of common two-dimensional shapes. Right Triangle Calculator: Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
How to use the Euclidean Distance Calculator
- Type point A, for example 1, 2, 3.
- Type point B, for example 4, 6, 3.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Point A | Text, such as 1, 2, 3 | — |
| Point B | Text, such as 4, 6, 3 | — |
Formula and method
Formula
d = √Σ(aᵢ − bᵢ)²
IEEE-754 double precision; shown to at most 6 decimal places.
Square root of the summed squared coordinate differences.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Point A = 1, 2, 3; Point B = 4, 6, 3.
| Result | Value |
|---|---|
| Distance | 5 |
Example 2
Inputs: Point A = 0 0; Point B = 1 1.
| Result | Value |
|---|---|
| Distance | 1.414214 |
The first line is the headline result (distance). IEEE-754 double precision; shown to at most 6 decimal places.
Any dimension
The formula extends the Pythagorean theorem to as many dimensions as you like. In machine learning it measures the similarity of feature vectors; smaller distance means more similar.
Alternatives
Manhattan distance sums absolute differences (7 here), which suits grid movement. Euclidean distance is the straight-line 'as the crow flies' measure. Points must have equal dimension.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Point B: entering "1, 2, 3" shows "Both points need the same number of coordinates." beside that field and no result.
Choosing between the Euclidean Distance Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Distance Between Two Points Calculator](/math/distance-between-two-points/) | x₁, y₁ and x₂ | Distance |
| [Area Calculator](/math/area-calculator/) | shape and a | Area |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Pythagorean Theorem Calculator](/math/pythagorean-theorem-calculator/) | find, leg a and other side | Hypotenuse c |
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Both points need the same number of coordinates
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for point B to confirm this.
Frequently asked questions
What is the Manhattan distance here?
The Manhattan distance between (1, 2, 3) and (4, 6, 3) is |3| + |4| + |0| = 7, against 5 for the straight line.
Does scale matter?
Yes, normalise features of different units.
What is the distance between identical points?
Zero, because every coordinate difference is zero.
What does the Euclidean Distance Calculator calculate?
Distance between two points with any number of coordinates. It returns Distance.
What formula does the Euclidean Distance Calculator use?
d = √Σ(aᵢ − bᵢ)². Square root of the summed squared coordinate differences.
What do I need to enter in the euclidean Distance Calculator?
You enter Point A and Point B.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Both points need the same number of coordinates.
What is an example of a euclidean Distance result?
With point A 1, 2, 3 and point B 4, 6, 3, the distance is 5.
Related calculators
- Distance Between Two Points Calculator — Straight-line distance and midpoint between two points in the plane.
- Area Calculator — Area of common two-dimensional shapes.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Pythagorean Theorem Calculator — Find the hypotenuse or a missing leg of a right triangle.
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- Circle Calculator — Radius, diameter, circumference and area of a circle from any one of them.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.