Distance Formula Calculator - 2D and 3D Distance
Calculate the Euclidean distance between two points in 2D or 3D space using the distance formula, with the full working shown.
Distance Formula Calculator - 2D and 3D Distance
Calculate the Euclidean distance between two points in 2D or 3D space using the distance formula, with the full working shown.
Point 1
Point 2
Load a quick example:
About the Distance Formula Calculator
The distance formula is one of the most widely used results in coordinate geometry. It gives the straight-line, or Euclidean, distance between any two points in the plane or in space. In two dimensions, the formula is d = √((x₂−x₁)² + (y₂−y₁)²); in three dimensions it extends to d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). Both formulas are direct applications of the Pythagorean theorem: the horizontal and vertical (and depth) differences form the legs of a right triangle, and the distance between the points is the hypotenuse.
The 2D formula appears throughout elementary coordinate geometry. Any time you need the length of a line segment joining two known points — the side of a triangle, the radius of a circle given its center and a boundary point, the distance between two cities on a map — the distance formula provides the answer in a single calculation. The 3D version is equally important in solid geometry, computer graphics, robotics, and physics, where positions in space are represented as triples (x, y, z).
A useful special case is the distance from a point to the origin. Setting (x₁, y₁) = (0, 0) reduces the 2D formula to d = √(x₂² + y₂²), which is also the formula for the magnitude (length) of the vector (x₂, y₂). This connection between distance and vector magnitude is central to linear algebra: the Euclidean norm of a vector is exactly the distance from the tip of the vector to the origin.
Computer science and data science rely heavily on Euclidean distance. In machine learning, the k-nearest neighbors algorithm classifies data points based on their distance to labeled examples. In clustering algorithms like k-means, points are assigned to the cluster whose center is nearest in Euclidean distance. In image processing, the Euclidean distance between pixel color values measures color similarity. In computer graphics, distance calculations underlie collision detection, ray-casting, and shading models.
For very large or very small coordinates, the calculator uses standard floating-point arithmetic, giving results accurate to at least ten significant digits. The formula is symmetric — swapping the two points gives the same distance — so the order in which you enter them does not matter. Enter any two points in 2D or 3D and the distance formula calculator returns the exact Euclidean distance along with the formula so you can verify the calculation step by step.
Distance Formula Examples
Worked examples showing 2D and 3D distance calculations with full explanations.
| Points | Distance | Explanation |
|---|---|---|
| 2D: (0,0) to (3,4) | 5 | d = √((3−0)²+(4−0)²) = √(9+16) = √25 = 5. This is the famous 3-4-5 right triangle. |
| 2D: (−1,2) to (2,6) | 5 | d = √((2−(−1))²+(6−2)²) = √(9+16) = √25 = 5. Another 3-4-5 triangle, shifted from the origin. |
| 3D: (0,0,0) to (1,1,1) | ≈ 1.732 | d = √(1+1+1) = √3 ≈ 1.732. This is the main diagonal of a unit cube. |
| 3D: (1,2,3) to (4,6,8) | ≈ 7.071 | d = √((3)²+(4)²+(5)²) = √(9+16+25) = √50 = 5√2 ≈ 7.071. |
How to Use the Distance Formula Calculator
- Choose the dimension — 2D for two-dimensional coordinates (x, y) or 3D for three-dimensional coordinates (x, y, z).
- Enter the coordinates of Point 1 (x₁, y₁, and optionally z₁) in the first group of fields.
- Enter the coordinates of Point 2 (x₂, y₂, and optionally z₂) in the second group of fields.
- Click Calculate Distance to see the Euclidean distance and the formula used.
- Use the quick-load buttons to see classic worked examples, or click Reset to clear all fields.
Distance Formula Calculator FAQ
What is the distance formula?
The distance formula calculates the straight-line (Euclidean) distance between two points. In 2D, d = √((x₂−x₁)²+(y₂−y₁)²). In 3D, d = √((x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²). Both formulas come directly from the Pythagorean theorem applied to the difference in each coordinate.
Why is the distance formula based on the Pythagorean theorem?
The horizontal difference (x₂−x₁) and vertical difference (y₂−y₁) form two legs of a right triangle, with the line segment between the points as the hypotenuse. The Pythagorean theorem a²+b²=c² gives the hypotenuse as √(a²+b²), which is exactly the distance formula. In 3D, there is a third leg (z₂−z₁) and the same logic extends to three dimensions.
Does the order of the points matter?
No. Because each coordinate difference is squared, (x₂−x₁)² = (x₁−x₂)². The distance from A to B equals the distance from B to A. You can enter the points in either order and get the same result.
Can I use the distance formula with negative coordinates?
Yes. Negative coordinates work exactly the same way. For example, the distance from (−3, −4) to (0, 0) is √(9+16) = 5. The subtraction handles negative values correctly, and squaring removes any sign issues.
What is the distance between two identical points?
Zero. If both points are the same, every difference (x₂−x₁), (y₂−y₁), and (z₂−z₁) equals zero, so the sum of squares is zero and the square root is zero. Geometrically, a point has zero distance from itself.
How is Euclidean distance different from other distance metrics?
Euclidean distance is the straight-line distance — the shortest path through open space. Other metrics include Manhattan distance (sum of absolute differences, like city blocks), Chebyshev distance (maximum absolute difference), and cosine similarity (angle between vectors). The distance formula always computes Euclidean distance, which is the most common metric in geometry and everyday applications.