Distance Between Two Points
Calculate the straight-line distance between two coordinate points using the distance formula.
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How to use this tool
- Enter x₁, y₁, x₂ and y₂ in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your distance and the full breakdown beneath it.
Find the straight-line (Euclidean) distance between two points using d = √((x₂−x₁)² + (y₂−y₁)²).
Formula
d = √((x₂ − x₁)² + (y₂ − y₁)²)
This is the Euclidean distance formula, derived from the Pythagorean theorem applied to the horizontal and vertical separations.
How it works
The calculator finds the horizontal difference (x₂ − x₁) and vertical difference (y₂ − y₁), squares each, sums them, and takes the square root — giving the straight-line (Euclidean) distance in the same units as the coordinates. The result is always non-negative and assumes a flat, two-dimensional plane.
Worked example
Worked example: distance from (0, 0) to (3, 4)
- Inputs: x₁ = 0, y₁ = 0, x₂ = 3, y₂ = 4.
- Horizontal difference: 3 − 0 = 3; square it: 9.
- Vertical difference: 4 − 0 = 4; square it: 16.
- Sum of squares: 9 + 16 = 25.
- Distance: √25 = 5.
Distance = 5
Common mistakes to avoid
- Forgetting to take the square root at the end, returning the squared distance instead of the actual distance.
- Confusing Manhattan distance (|x2-x1|+|y2-y1|) with Euclidean distance for grid-based problems.
- Assuming the formula works the same in 3D without adding the (z2-z1)² term under the square root.
Key terms
- Euclidean distance
- The straight-line distance between two points in a plane, computed with the distance formula.
- Coordinate plane
- A two-dimensional surface defined by perpendicular x- and y-axes used to locate points.
- Pythagorean theorem
- The rule a² + b² = c² for right triangles; the distance formula is a direct application of it.
- Cartesian coordinates
- An ordered pair (x, y) specifying a point's position relative to the origin of a coordinate plane.
Frequently asked questions
- What is the distance formula?
- The distance between (x₁,y₁) and (x₂,y₂) is d = √((x₂−x₁)² + (y₂−y₁)²). This follows from the Pythagorean theorem.