Algebra⏱ 4 min read

Distance Formula

d = √((x₂-x₁)² + (y₂-y₁)²)

What is the Distance Formula?

The distance formula finds the straight-line distance between two points on a coordinate plane. It's derived directly from the Pythagorean theorem: the horizontal gap (x₂−x₁) and vertical gap (y₂−y₁) between the points form the two legs of a right triangle, and the distance between the points is that triangle's hypotenuse.

The formula extends naturally to three dimensions by adding a third squared term for the z-coordinates: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²).

What Each Variable Means

DistanceThe straight-line length between the two points.

x₁, y₁

First pointThe coordinates of the starting point.

x₂, y₂

Second pointThe coordinates of the ending point.

When to Use It

Finding the straight-line distance between two points on a map or graph
Verifying whether three points form a particular type of triangle by comparing side lengths
As a building block for circle equations and other coordinate geometry problems

Where This Formula Comes From

Draw the two points on a coordinate plane

The horizontal and vertical differences between them form two legs of a right triangle, with the segment connecting the points as the hypotenuse.

Apply the Pythagorean theorem

The hypotenuse squared equals the sum of the legs squared.

d² = (x₂ - x₁)² + (y₂ - y₁)²

Take the square root of both sides

Solve for the distance itself.

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Step-by-Step Example

Problem: Find the distance between points (1, 2) and (4, 6).

Label the coordinates

Assign each point's x and y values.

x₁=1, y₁=2, x₂=4, y₂=6

Calculate the differences

Subtract the x-values and the y-values.

(x₂-x₁) = 3, (y₂-y₁) = 4

Square each difference

Square both results.

3² = 9, 4² = 16

Add and take the square root

Sum the squares, then take the square root.

d = √(9 + 16) = √25

✓

Answer: d = 5 units

Interactive Calculator

x₁y₁x₂y₂

Result will appear here

Common Mistakes

Mistake: Subtracting in the wrong order and losing track of which point is which.
Fix: The subtraction order doesn't matter for the final answer (both differences get squared, so signs cancel), but stay consistent — pick one point as (x₁,y₁) and keep it that way through the whole calculation.
Mistake: Forgetting to square each difference before adding.
Fix: You must square (x₂-x₁) and (y₂-y₁) individually before adding them — adding first and squaring the sum gives a different, wrong result.

Practice Questions

Find the distance between (0, 0) and (6, 8).
Hint: This is a 6-8-10 right triangle, a multiple of the 3-4-5 triangle.
Find the distance between (-2, 3) and (1, -1).

Frequently Asked Questions

Does the distance formula work in 3D?

Yes — add a third squared term for the z-coordinates: d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²).

Why is it related to the Pythagorean theorem?

Because the horizontal and vertical gaps between two points literally form the two legs of a right triangle, with the straight-line distance as the hypotenuse.