Find the Euclidean distance between two points in 2D or 3D — plus midpoint, slope, Manhattan and Chebyshev distances with step-by-step working.
Pick 2D or 3D and enter coordinates — distance updates live.
Euclidean distance
Midpoint
(2.5, 4)
average of endpoints
Slope (2D only)
1.333
Δy / Δx
Manhattan
7
|Δx| + |Δy|
Chebyshev
4
max(|Δx|, |Δy|)
Δx
3
x₂ − x₁
Δy
4
y₂ − y₁
Δz
—
z₂ − z₁
Angle (from +x)
53.13°
atan2(Δy, Δx)
| Metric | 2D formula | Use case |
|---|---|---|
| Euclidean | √(Δx² + Δy²) | geometry, straight-line distance |
| Manhattan (L¹) | |Δx| + |Δy| | city grids, taxi routes |
| Chebyshev (L∞) | max(|Δx|, |Δy|) | chessboard king's moves |
| Minkowski (Lᵖ) | (|Δx|ᵖ + |Δy|ᵖ)^(1/p) | general L-norm family |
| Midpoint | ((x₁+x₂)/2, (y₁+y₂)/2) | centre of segment |
| Slope | (y₂ − y₁)/(x₂ − x₁) | line steepness |
The Method
The Euclidean distance between two points is the straight-line length of the segment that joins them. It comes directly from the Pythagorean theorem: treat Δx and Δy as the legs of a right triangle, and the hypotenuse d = √(Δx² + Δy²) is the distance. In three dimensions the same trick repeats: d = √(Δx² + Δy² + Δz²). Other distance metrics — Manhattan, Chebyshev, Minkowski — change how the coordinate differences are combined, but every one of them assigns a non-negative number to every pair of points and equals zero iff the points coincide.
Working for current points
About This Tool
A distance calculator finds the straight-line (Euclidean) length between two points in two or three dimensions. Enter the coordinates of Point 1 and Point 2 and the calculator returns the distance, the midpoint, the slope (2D only), the per-axis differences Δx, Δy, Δz, the angle from the +x axis, and two non-Euclidean cousins — Manhattan and Chebyshev distance.
The Euclidean formula d = √(Δx² + Δy²) follows from the Pythagorean theorem: Δx and Δy form the legs of a right triangle whose hypotenuse is the distance you want. In three dimensions the same idea extends naturally to d = √(Δx² + Δy² + Δz²). This calculator evaluates the formula in IEEE 754 double precision — accurate to about 15 significant digits, far beyond the precision of any real-world measurement.
Beyond the textbook Euclidean metric, the calculator reports two important alternatives. Manhattan distance (|Δx| + |Δy|) measures distance along axis-aligned moves — the path a taxi takes through a city grid. Chebyshev distance (max(|Δx|, |Δy|)) measures the minimum number of moves for a chess king. Both are Lᵖ-norm generalisations of the same idea and show up across machine learning, robotics, computer graphics, and pathfinding algorithms.
Use this free distance calculator for geometry homework, computer graphics, GIS, route-planning sketches, or any time you need a quick read on how far apart two points are. All computation runs locally — no sign-up, no tracking, no data sent to any server.
2D and 3D Distance
Toggle between (x, y) and (x, y, z) — formula and stats update automatically.
Midpoint & Slope
Reports midpoint coordinates and (in 2D) line slope from Δy / Δx.
Manhattan & Chebyshev
Two non-Euclidean metrics for grid-based and chessboard-style problems.
Live Diagram
SVG plot shows both points, segment, and midpoint marker.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Step-by-Step Working
Each formula substitution shown — ideal for homework or revision.
From points to full geometry in seconds.
Use the tabs to toggle between 2D (x, y) and 3D (x, y, z). The z-input appears only when 3D is selected.
Type the coordinates of the first point — negative numbers are fine, just use the minus sign.
Type the second point. The distance, midpoint, and slope all update the moment you change a value.
The headline shows the Euclidean distance. Stat tiles include Manhattan, Chebyshev, midpoint, slope, Δx, Δy, Δz and the angle.
In 2D mode the SVG plots both points, the segment between them, and the midpoint — useful for sanity-checking your input.
The formula card shows the substitution (Δx, Δy, …) and final answer — ideal for assignment write-up.
Everything you need to know about distance between points.
In 2D: d = √((x₂ − x₁)² + (y₂ − y₁)²). In 3D add (z₂ − z₁)² inside the root. Both come from the Pythagorean theorem — the legs are the per-axis differences and the hypotenuse is the distance.
Manhattan distance (also called L¹ or taxicab distance) is the sum of absolute differences along each axis: |Δx| + |Δy| (+ |Δz|). It models the path of a taxi in a city grid where you can only move along streets. It's never smaller than the Euclidean distance.
Chebyshev distance (also called L∞ or chessboard distance) is the maximum absolute difference along any axis: max(|Δx|, |Δy|). It models the minimum number of moves a chess king needs since each move can change x and y by 1 simultaneously.
The midpoint is the average of the two endpoints: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). In 3D add the z-average. It's equidistant from both endpoints and lies on the segment joining them.
Slope measures the line's steepness — how much y changes per unit of x — and is computed as (y₂ − y₁) / (x₂ − x₁). Distance measures how far apart the points are along the segment. Two points can be the same distance apart but lie on lines of very different slope.
Distance is a non-negative scalar — just a length. Displacement is a vector that includes direction. Two points always have the same distance no matter which is "first", but their displacement reverses sign when you swap them.
The Minkowski distance with parameter p is (|Δx|ᵖ + |Δy|ᵖ)^(1/p). It generalises every standard metric: p = 1 is Manhattan, p = 2 is Euclidean, p → ∞ is Chebyshev. It's widely used in machine learning for k-nearest-neighbour distance metrics.
Common causes: (1) mixing up signs — remember Δx² and Δy² are always positive; (2) swapping x and y — the order of subtraction doesn't matter under the square because of the squaring; (3) forgetting the square root. The calculator shows Δx and Δy separately so you can verify your inputs.
Yes — just type the negative sign. The formula works for all real coordinates, in any quadrant or octant. The distance is always non-negative no matter which sign the coordinates have.