Euclidean Distance Calculator - L2 Norm and Squared Distance

The euclidean distance calculator returns the straight-line distance between two points of any dimension by summing the squared per-axis differences and taking the square root. Type the coordinates of point A and point B as comma-, semicolon-, or space-separated numbers and the page returns d, d squared, the max absolute difference, and the dimension count. The same formula covers 2D point pairs, 3D coordinates, and high-dimensional vectors used in machine learning, color matching, and similarity search.

• • Planar geometry homework: Check a coordinate-geometry answer for two points on a plane, including negative and decimal coordinates.
• • 3D distance for design and physics: Compute the distance between two points in 3D space, such as a toolpath, a CAD part, or two sensor readings.
• • High-dimensional machine learning: Compute the L2 norm and squared distance between two feature vectors in k-NN, k-means clustering, or similarity scoring.
• • Color and pixel difference: Measure the distance between two RGB or RGBA colors, where each channel is one dimension of the input.

The two-point inputs accept commas, semicolons, or whitespace between numbers, and the dimension count is whatever the user types, so the same form works for a quick 2D check and a longer n-dimensional feature vector.

The squared distance is shown alongside d, so clustering and nearest-neighbor code that compares d squared first to skip a square root can be validated from the same run.

If you only have a single point pair on a plane, the 2D Distance Calculator page is the lighter-weight version of the same arithmetic for two x-y coordinates.

The page parses each point into a numeric vector, builds the element-wise difference, sums the squared differences, and takes the square root. All four supporting outputs come from the same per-axis differences in a single pass.

• x1, x2, ..., xn: Coordinates of point A. The list must contain the same number of entries as point B.
• y1, y2, ..., yn: Coordinates of point B in the same order as point A.
• Δi = yi - xi: Per-axis signed difference. Squaring removes the sign so the order of the two points does not change d.
• d = sqrt(sum of Δi^2): The Euclidean distance, also called the L2 norm of the difference vector. Always non-negative.

The squared sum avoids the cost of a square root on every comparison, which is why k-means, k-NN, and many nearest-neighbor implementations compare d squared instead of d.

The max absolute difference is a useful lower bound: it tells you the largest single-axis contribution. If the max equals d itself, the per-axis differences line up in the same direction.

According to Wolfram MathWorld, the n-dimensional Euclidean distance is the square root of the sum of the squared coordinate differences, exactly the formula this page evaluates for user-supplied dimensions.

According to Khan Academy, the two-dimensional distance formula is a direct application of the Pythagorean theorem, so right triangles with integer legs that land on a whole-number hypotenuse give a clean integer d.

The 2D case of this formula is the Pythagorean theorem, and the Pythagorean Theorem Solver page solves for any single side of a right triangle when you already know the other two.

Four short ideas explain what the formula computes and how the supporting outputs connect to it.

The squared distance and the max absolute difference are both cheap to compute from the same per-axis differences, so the four outputs together cover the cases a script reaches for.

If you mainly care about the L2 norm of a single vector from the origin, the dedicated vector magnitude page returns the same arithmetic when the second point is the origin.

When one of the two points is the origin, the distance is the magnitude of the other point, and the Vector Magnitude Calculator page runs the same L2-norm arithmetic on a single vector.

Four quick steps cover the common cases from a 2D textbook example to a high-dimensional vector.

1. 1 Type the coordinates of point A: Write the first point as a list of numbers separated by commas, semicolons, or spaces. Use at least two coordinates and at most ten.
2. 2 Type the coordinates of point B: Repeat the format for the second point, with the same number of coordinates as point A.
3. 3 Read the Euclidean distance: The primary output updates as you type. d is the L2 norm of the difference vector and is always non-negative, even when one or both points contain negative coordinates.
4. 4 Check the squared distance, max difference, and dimension count: Use the squared distance for clustering, the max absolute difference as a single-axis lower bound, and the dimension count to confirm both lists are the same length.

Try point A = (1, 2, 3) and point B = (3, 5, 9). The page returns d = 7, d squared = 49, max|Δ| = 6, and dimensions = 3. The squared sum is 49, so d is a clean integer and the max difference is the z-axis jump from 3 to 9.

For a quick spatial-only check, the 3D Distance Calculator page uses the same formula on x, y, and z coordinates and adds the 3D midpoint and space diagonal in one pass.

These are the practical reasons the page is worth opening, even when you already know the formula.

• • Works in 2D, 3D, and higher dimensions from one form: Type two points of any matching length, from a quick 2D check to a 10-dimensional feature vector, and the same form returns the distance.
• • Returns the squared distance alongside d: Clustering and nearest-neighbor code compares d squared first to skip a square root. The page shows both side by side, so you can validate both at once.
• • Flexible coordinate input format: Commas, semicolons, or spaces between numbers all work. Negative and decimal coordinates are accepted without preprocessing.
• • Clear error messages for bad input: If the two points have different lengths, contain a non-numeric token, or have fewer than two coordinates, the page says so directly instead of silently returning NaN.
• • Connects to the rest of analytic geometry: Use the 2D page for planar problems, the 3D page for spatial ones, and the vector magnitude page for single-vector lengths, all from the same category.

The most common reason to reach for the page is to confirm what a script will produce, especially around the squared-distance value that clustering and similarity code uses, or to check the math behind a high-dimensional pipeline before shipping it.

When the result needs to switch units, the Distance Converter page translates the straight-line distance into meters, feet, kilometers, miles, and other length units without re-running the formula.

The formula is fixed, but the inputs and the way the result is read change with a few factors.

• • The page assumes a flat Euclidean space. It does not account for great-circle distance on a sphere or Manhattan distance along a road network.
• • The result is the straight-line distance, not the path length. For two points on a curve, the straight-line distance is a lower bound on the path length, not the path length itself.
• • Inputs beyond ten coordinates are rejected to keep the form readable. For longer vectors, paste them into a script or split them into batches.

If d looks small, the per-axis differences are likely small in absolute terms; if d looks large, the max absolute difference usually points at one dominant axis, and the squared-distance output makes that contribution visible before you take the square root.

According to Wikipedia, the Euclidean distance between two points in Euclidean space is the length of the line segment between them, given by the L2 norm of the difference vector, and is widely used in machine learning, clustering, and similarity search.

If the per-axis differences for a 2D point pair come out as integers and d is also an integer, the Pythagorean Triples Calculator page identifies the Pythagorean triple those coordinates describe.