2D Distance Calculator - Euclidean Formula and Steps

A 2D distance calculator finds the straight-line distance between two points on a coordinate plane using the Euclidean distance formula. It is the everyday analytic-geometry tool for the question, "how far apart are (x1, y1) and (x2, y2)?" You type the four coordinates, the page shows the distance, the midpoint, the slope of the line through the points, and the step-by-step math, so you can both check your homework and trust the answer.

• • Math homework and analytic geometry: Confirm a problem that asks for the distance between two points, the midpoint, or the slope of the line through them, including cases that come out to neat Pythagorean triples.
• • Coordinate geometry and graphing: Plot two points on a coordinate plane, then read the distance and the midpoint in one pass without re-entering numbers into a separate tool.
• • Mapping and screen coordinates: Measure the pixel or grid distance between two on-screen locations when you are working out positions for a layout, sprite, or design grid.
• • Physics and engineering preliminaries: Compute the magnitude of a 2D displacement vector, which is just the Euclidean distance from the start point to the end point, before moving on to forces or velocities.

The 2D distance calculator is intentionally narrow: it answers the Euclidean distance question for the planar case, where every point has exactly two coordinates. If you need a 3D magnitude, or you want to change the unit of the result from units to meters, feet, miles, or pixels, pair this page with one of the peer calculators below.

Because the page shows the dx, dy, and the squared distance alongside the final answer, you can see exactly which arithmetic produced the result. That makes it useful as a teaching aid, not just a black-box answer machine.

If the answer is in generic "units" and you want meters, feet, miles, or pixels, the Distance Converter page changes the unit of the result.

The page implements the standard Euclidean distance formula in 2D. It takes the two coordinate inputs, builds the horizontal and vertical differences, squares and sums them, and takes the square root.

• x1, y1: Coordinates of the first point on the plane.
• x2, y2: Coordinates of the second point on the plane.
• Δx = x2 − x1: Horizontal change between the two points, signed.
• Δy = y2 − y1: Vertical change between the two points, signed.
• d = sqrt(Δx² + Δy²): The Euclidean distance, which is always non-negative.

The same formula works for negative coordinates and for decimal coordinates. Squaring removes the sign of Δx and Δy, so the distance is always non-negative no matter which point you call first.

According to Khan Academy, the distance between two points (x1, y1) and (x2, y2) on a coordinate plane is computed as the square root of (x2 - x1)^2 + (y2 - y1)^2, which is the Pythagorean theorem applied to the right triangle formed by the two points.

If you would rather see the magnitude of a 2D vector that starts at the origin, the Vector Magnitude Calculator page runs the same arithmetic on a single vector instead of two points.

Four ideas explain why the formula is what it is and what the result really means on the plane.

These four concepts are the building blocks of the rest of analytic geometry. If you have the distance, the midpoint, and the slope, you can reconstruct the line through the two points in almost any form a textbook will ask for.

The 3-4-5 example is a special case of a Pythagorean triple, and if you generate many pairs of points and want a quick way to spot the ones that come out to integers, the dedicated Pythagorean-triples page is the next click.

The 3-4-5 example is a special case of a Pythagorean triple, and if you generate many pairs of points and want a quick way to spot the ones that come out to integers, the Pythagorean Triples Calculator page does that directly.

Five short steps cover every common case, from a clean textbook example to negative coordinates.

1. 1 Enter x₁ and y₁: Type the coordinates of the first point. The default is (0, 0); change both values if your point is somewhere else on the plane.
2. 2 Enter x₂ and y₂: Type the coordinates of the second point. The default is (3, 4) so the example starts as the classic 3-4-5 right triangle.
3. 3 Read the distance: The primary output is the Euclidean distance d, updated as you type. It is the same number whether you swap the two points or not.
4. 4 Check the supporting values: Look at Δx, Δy, the squared distance, and the midpoint to confirm the arithmetic. The slope is shown as a number, or as a clear marker when the line is vertical.
5. 5 Reset or change units if needed: Click Reset to return to the example. If the answer is in generic "units" and you want meters, feet, or pixels, open the Distance Converter in a new tab.

Try the points (−2, −3) and (3, 9). The calculator gives Δx = 5, Δy = 12, d = 13, midpoint (0.5, 3), and slope 12 / 5. The result 13 confirms that (5, 12, 13) is another Pythagorean triple, this time with negative coordinates on the first point.

Once you have the distance, midpoint, and slope, the Right Triangle Calculator page can fill in the missing leg, hypotenuse, or angle of the right triangle those values describe.

These benefits matter most when you are working a problem by hand and need a quick, trustworthy check.

• • Skip the arithmetic mistakes: Manual distance problems are easy to get wrong on the squaring step. The calculator handles the squaring and square root so you can focus on setting up the problem.
• • See the full step-by-step math: The page shows Δx, Δy, the squared distance, and the final d. That makes it a good way to check your own work, not just a way to get a number.
• • Get the midpoint and slope for free: You often need the midpoint of a segment or the slope of the line through two points at the same time as the distance. The calculator returns all three from the same four inputs.
• • Handle any sign of coordinate: Negative coordinates, decimal coordinates, and points far from the origin all use the same formula. The page does not require you to pre-process the numbers.
• • Connect to the rest of analytic geometry: If your next step is to draw the right triangle, find a Pythagorean triple, or convert the result to a different unit, the page links out to the relevant peer calculator in the same category.

The page is most useful as a check, not as a replacement for understanding the formula. Use it to confirm a homework answer, sanity-check a graphics calculation, or pre-validate a coordinate pair before you hand it to a longer script.

If you spend more time on the problem in front of you than on the arithmetic, the supporting outputs give you a second way to be sure the four coordinates you entered are the ones you meant to type.

If you want to plot the two points and the line through them on a labeled grid, the Coordinate Plane Calculator page draws the picture that the numbers describe.

The formula is the same in every case, but a few factors change how the result should be read.

• • This page is the planar, 2D case only. For 3D points (x, y, z), the formula needs an extra z term; the dedicated 3D vector magnitude page handles that directly.
• • The calculator assumes a flat Euclidean plane. It does not account for distance on the surface of a sphere (great-circle distance) or along a road network (Manhattan distance).
• • The result is the straight-line distance, not the path length. If the points lie on a curve, the straight-line distance is a lower bound on the path length, not the path length itself.

According to Wolfram MathWorld, the Euclidean distance between two points in two-dimensional space is given by d = sqrt((x2 - x1)^2 + (y2 - y1)^2), which generalizes the Pythagorean theorem to coordinate points.

When the two points sit on a known curve and the path length between them matters more than the straight-line distance, the Arc Length Calculator page integrates the distance along that curve.