Distance Between Two Points Calculator

The distance between two points calculator measures the straight-line (Euclidean) distance between any two points on a coordinate plane. Type in the x and y values of each point and the result appears instantly, shown as an exact radical form when the answer is not a whole number and as a decimal approximation.

The distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²] comes directly from the Pythagorean theorem, applied to the horizontal and vertical gaps between the two points.

Who Uses It

Geometry & Navigation

Measure segment lengths, gaps on a map grid, or the shortest path between two points on a plane.

Graphics & Games

Work out spacing between objects, collision ranges, or how far a sprite must travel on screen.

Homework & Study

Check your answers and follow the worked steps to learn how the distance formula is applied.

How to Find the Distance Between Two Points

Enter the First Point

Type the x-coordinate (x₁) and y-coordinate (y₁) of the first point into the input fields.

Enter the Second Point

Type the x-coordinate (x₂) and y-coordinate (y₂) of the second point. Decimal and negative values are both accepted.

Read the Result

The distance updates automatically as you type — shown in exact form (with √) and as a decimal, along with the Δx and Δy components.

Review the Steps

Open the step-by-step solution to see every calculation, and check the graph plotting both points and the segment between them.

Private by design: every calculation runs locally in your browser — no coordinates are sent to a server.

What the Distance Calculator Shows

The Distance Formula, Step by Step

The distance between point P₁(x₁, y₁) and point P₂(x₂, y₂) is the length of the hypotenuse of a right triangle whose legs are the horizontal change (Δx) and the vertical change (Δy).

Step	Expression
Horizontal change (Δx)	`Δx = x₂ − x₁`
Vertical change (Δy)	`Δy = y₂ − y₁`
Distance (d)	`d = √[(Δx)² + (Δy)²]`

Worked example: for points (0, 0) and (3, 4): Δx = 3, Δy = 4, so d = √(3² + 4²) = √(9 + 16) = √25 = 5.

Results You Get

Exact & Decimal Forms

Shows the simplified exact radical (such as √2 or 5√2) next to a decimal approximation.

Step-by-Step Solution

A collapsible breakdown of every calculation so you can follow exactly how the answer is reached.

Visual Graph

An interactive graph plots both points, labels P₁ and P₂, and draws the segment connecting them.

Δx and Δy Components

Shows the horizontal (Δx) and vertical (Δy) changes that form the two legs of the right triangle.

Reference results: (0, 0) and (3, 4) → 5; (1, 2) and (4, 6) → 5; (0, 0) and (1, 1) → √2 ≈ 1.414.

Frequently Asked Questions

What is the distance formula?

The distance formula is d = √[(x₂ − x₁)² + (y₂ − y₁)²]. It gives the straight-line distance between two points on a coordinate plane by treating the horizontal and vertical gaps as the two legs of a right triangle and the distance as its hypotenuse.

How do you find the distance between two points?

Label the points (x₁, y₁) and (x₂, y₂), subtract to get Δx = x₂ − x₁ and Δy = y₂ − y₁, square both differences, add them, then take the square root. This calculator does every step for you and shows the working.

Is the distance formula the same as the Pythagorean theorem?

They are the same idea. The Pythagorean theorem states a² + b² = c²; the distance formula simply names the two legs Δx and Δy, so the distance c equals √(Δx² + Δy²).

What does the exact (radical) form mean?

When the distance is not a whole number, the calculator shows it as a simplified square root — such as √2 or 5√2 — instead of only a rounded decimal. This keeps the value exact and matches how answers are written in most textbooks.

Can I use negative or decimal coordinates?

Yes. Any real numbers work, including negative values and decimals. Because the differences are squared, the sign of Δx and Δy never changes the final distance.

Does this calculate 3D distance?

This tool works with 2D coordinates on a plane. For three dimensions you would extend the formula with a z-term: d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].