Distance Formula Calculator: How to Calculate Distance
Use the distance formula to find the straight-line distance between two points on a two-dimensional Cartesian coordinate plane.
Quick answer: A distance formula calculator applies the coordinate-distance equation to find the straight-line distance between two points; geographic coordinates need a different model.
For:
Point 1 = (x₁, y₁)
Point 2 = (x₂, y₂)
the formula is:
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]
What Does the Distance Formula Calculate?
The distance formula calculates the Euclidean distance between two Cartesian points.
If the points are:
A = (x₁, y₁)
and:
B = (x₂, y₂)
their straight-line separation is:
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]
This is the length of the line segment connecting the two points on a flat coordinate plane.
For the broader theory and related distance equations, see the Distance Formula guide.
What Values Go Into the Distance Formula?
A standard two-dimensional distance calculation requires four coordinate values:
- x₁
- y₁
- x₂
- y₂
These form two ordered pairs:
Point 1 = (x₁, y₁)
Point 2 = (x₂, y₂)
For example:
Point 1 = (1, 2)
Point 2 = (4, 6)
The four values are substituted into the same Euclidean distance equation.
What Is the Distance Formula?
The formula for two points on a Cartesian plane is:
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]
Where:
| Symbol | Meaning |
|---|---|
| x₁ | x-coordinate of Point 1 |
| y₁ | y-coordinate of Point 1 |
| x₂ | x-coordinate of Point 2 |
| y₂ | y-coordinate of Point 2 |
| Distance | Straight-line distance between the points |
The formula comes from the Pythagorean theorem.
Our Euclidean Distance Formula guide explains the mathematical derivation in detail.
How Do You Calculate Distance Between Two Points?
The calculation follows five steps.
1. Find the difference between the x-coordinates
Δx = x₂ − x₁
2. Find the difference between the y-coordinates
Δy = y₂ − y₁
3. Square both differences
Δx²
and:
Δy²
4. Add the squared differences
Δx² + Δy²
5. Take the square root
Distance = √(Δx² + Δy²)
The result is the straight-line Euclidean distance between the two points.
Distance Formula Example
Find the distance between:
A = (1, 2)
B = (4, 6)
Start with:
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substitute the values:
Distance = √[(4 − 1)² + (6 − 2)²]
Calculate the differences:
Distance = √(3² + 4²)
Square them:
Distance = √(9 + 16)
Add:
Distance = √25
Final result:
Distance = 5 units
Example With Negative Coordinates
Suppose:
A = (−3, −1)
B = (2, 3)
Substitute the values:
Distance = √[(2 − (−3))² + (3 − (−1))²]
Simplify:
Distance = √(5² + 4²)
Distance = √(25 + 16)
Distance = √41
Distance ≈ 6.40 units
Negative coordinates do not require a different formula.
The signs matter while finding the differences, but the squared differences are nonnegative.
Example With Decimal Coordinates
Suppose:
A = (1.5, 2.5)
B = (5.5, 5.5)
Calculate:
Distance = √[(5.5 − 1.5)² + (5.5 − 2.5)²]
Distance = √(4² + 3²)
Distance = √(16 + 9)
Distance = √25
Distance = 5 units
The Euclidean distance formula works with decimal values as well as integers.
Why Does the Distance Formula Work?
The coordinate differences form the legs of a right triangle.
The horizontal leg has length:
|x₂ − x₁|
The vertical leg has length:
|y₂ − y₁|
The distance between the original points is the hypotenuse.
Using the Pythagorean theorem:
c² = a² + b²
gives:
Distance² = (x₂ − x₁)² + (y₂ − y₁)²
Taking the square root gives:
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]
If you want a more detailed explanation of the geometry, read Distance Between Two Points.
What Happens If the Points Are the Same?
The distance is 0.
For:
A = (4, 7)
B = (4, 7)
calculate:
Distance = √[(4 − 4)² + (7 − 7)²]
Distance = √(0 + 0)
Distance = 0
Two identical coordinate pairs represent the same point.
Can the Distance Be Negative?
No. Euclidean distance is always nonnegative.
A coordinate difference can be negative, such as:
2 − 6 = −4
but:
(−4)² = 16
Squaring the coordinate differences prevents the final distance from becoming negative.
Therefore:
Distance ≥ 0
Does Point Order Matter?
No.
The distance from Point A to Point B is the same as the distance from Point B to Point A.
For example:
5 − 2 = 3
while reversing the order gives:
2 − 5 = −3
But:
3² = (−3)² = 9
So:
Distance(A, B) = Distance(B, A)
This property is known as symmetry.
What If the Points Have the Same x-Coordinate?
If both points have the same x-coordinate, the distance is purely vertical.
For example:
A = (3, 2)
B = (3, 9)
Then:
Distance = √[(3 − 3)² + (9 − 2)²]
Distance = √(0 + 7²)
Distance = 7 units
The special case can be written as:
Distance = |y₂ − y₁|
What If the Points Have the Same y-Coordinate?
If both points have the same y-coordinate, the distance is purely horizontal.
For:
A = (2, 5)
B = (10, 5)
calculate:
Distance = √[(10 − 2)² + (5 − 5)²]
Distance = √(8² + 0)
Distance = 8 units
This can also be written as:
Distance = |x₂ − x₁|
What Units Does the Result Use?
The result uses the same underlying unit as the coordinate system.
For example:
- coordinates in meters produce distance in meters;
- coordinates in feet produce distance in feet;
- coordinates in centimeters produce distance in centimeters;
- coordinates without a physical unit produce a result in generic units.
If:
A = (0, 0)
and:
B = (3, 4)
with no measurement unit specified, the result should be written as:
5 units
Both coordinate axes should use compatible units.
How Does 2D Distance Differ From 3D Distance?
The examples on this page calculate two-dimensional distance from x and y coordinates. A three-dimensional point also has a z-coordinate.
For three-dimensional points, add the squared difference between the z-coordinates before taking the square root. The example below illustrates the mathematics; this page does not provide an interactive 3D calculator.
The 3D formula itself is:
Distance = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
For example:
A = (1, 2, 3)
B = (5, 5, 6)
gives:
Distance = √(4² + 3² + 3²)
Distance = √34
Distance ≈ 5.83 units
Can You Use This Formula for Latitude and Longitude?
Not for ordinary geographic distance across Earth’s surface.
The Cartesian distance formula assumes a flat Euclidean coordinate system.
Latitude and longitude are angular coordinates on a curved Earth.
For example, you should not simply calculate:
√[(longitude₂ − longitude₁)² + (latitude₂ − latitude₁)²]
and interpret the result as miles.
For geographic coordinates, use the Distance Calculator, which applies a spherical Haversine calculation.
The mathematical method is explained in our Haversine Formula guide.
Cartesian Distance Formula vs Geographic Distance Calculator
The two tools solve different problems.
| Feature | Cartesian Distance Formula | Geographic Distance Calculator |
|---|---|---|
| Coordinate system | Cartesian | Geographic |
| Typical inputs | x and y | Latitude and longitude |
| Geometry | Flat | Spherical approximation |
| Main method | Euclidean formula | Haversine formula |
| Typical result | Coordinate units | Miles, kilometers, nautical miles |
Use the worked formula on this page for geometry and Cartesian coordinates.
Use the Distance Calculator for latitude and longitude.
When Should You Use the Distance Formula?
Use the Euclidean distance formula for problems involving:
- coordinate geometry;
- graph points;
- Cartesian coordinates;
- x/y positions;
- engineering layouts using compatible planar units;
- computer graphics coordinates;
- classroom geometry problems;
- other flat coordinate systems.
The key requirement is that the points belong to a coordinate system where straight-line Euclidean geometry is appropriate.
When Should You Use Another Distance Method?
Use another method when the problem does not involve ordinary Cartesian coordinates.
Geographic Coordinates
Use the Distance Calculator for latitude and longitude.
Road Travel
Use the Driving Distance Calculator for mileage along roads.
Street Addresses
Use the Distance Between Addresses Calculator.
Map Points
Use the Map Distance Calculator when you want to select points visually.
Different distance tools exist because “distance” can describe different geometric and travel relationships.
Common Distance Formula Mistakes
Mixing x and y Values
Correct:
x₂ − x₁
and:
y₂ − y₁
Do not subtract an x-coordinate from a y-coordinate.
Mishandling Negative Coordinates
Use parentheses.
For:
x₁ = −3
and:
x₂ = 2
calculate:
2 − (−3) = 5
Forgetting to Square Both Differences
Both coordinate differences must be squared.
Forgetting the Square Root
If:
Distance² = 25
then:
Distance = 5
not 25.
Rounding Too Early
Keep intermediate values exact where practical and round the final answer.
Using the Wrong Coordinate Model
Do not use the flat Euclidean formula for geographic latitude and longitude unless the coordinates have first been transformed into an appropriate planar system.
Distance Formula Quick Reference
| Situation | Formula |
|---|---|
| Number line | Distance = |x₂ − x₁| |
| 2D Cartesian points | Distance = √[(x₂ − x₁)² + (y₂ − y₁)²] |
| 3D Cartesian points | Distance = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²] |
| Latitude/longitude | Use a geographic distance formula |
For a complete comparison of these equations, see the Distance Formula hub.
Frequently Asked Questions
What does the distance formula calculate?
The distance formula finds the straight-line Euclidean distance between Cartesian coordinate points.
For 2D coordinates:
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]
How do you calculate distance between two coordinate points?
Subtract the x-coordinates and y-coordinates separately, square both differences, add them, and take the square root.
What is the distance between (0, 0) and (3, 4)?
5 units.
Calculation:
Distance = √(3² + 4²)
Distance = √25
Distance = 5
Can the distance formula use negative coordinates?
Yes. The Euclidean formula works with negative coordinates because coordinate differences are squared.
Does the order of the points affect the result?
No. Swapping Point 1 and Point 2 changes the signs of the coordinate differences but not their squares, so the distance remains the same.
Can I use this formula for GPS coordinates?
Use a geographic distance calculator for raw GPS latitude and longitude. The ordinary Euclidean formula assumes flat Cartesian coordinates.
Is the distance formula the same as the Pythagorean theorem?
The 2D distance formula is derived from the Pythagorean theorem. The horizontal and vertical coordinate differences become the two legs of a right triangle, while the distance is the hypotenuse.
