Distance Calculator Guide: Distance Formula in 2D and 3D (2026)
Quick Answer
- *2D distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)
- *3D distance: add a z-term: d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
- *Derived from the Pythagorean theorem — distance is the hypotenuse of a right triangle.
- *Manhattan distance measures grid-based paths: d = |x₂ − x₁| + |y₂ − y₁|
The Distance Formula in 2D
The distance between two points (x₁, y₁) and (x₂, y₂) in a plane is:
d = √((x₂ − x₁)² + (y₂ − y₁)²)
This is the Pythagorean theorem in disguise. The horizontal distance (x₂ − x₁) and vertical distance (y₂ − y₁) form two legs of a right triangle. The straight-line distance between the points is the hypotenuse.
Worked Example
Find the distance between (1, 2) and (4, 6):
- d = √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5
The Distance Formula in 3D
Add the z-dimension:
d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
Distance between (1, 2, 3) and (5, 6, 7): d = √(16 + 16 + 16) = √48 ≈ 6.93
Manhattan Distance
Also called taxicab distance or L1 norm, Manhattan distance measures the path you'd travel along a grid — no diagonal shortcuts:
d = |x₂ − x₁| + |y₂ − y₁|
Between (1, 2) and (4, 6): d = |4 − 1| + |6 − 2| = 3 + 4 = 7 (compared to Euclidean distance of 5). Manhattan distance is always ≥ Euclidean distance.
When to Use Each Distance
| Distance Type | Formula | Best For |
|---|---|---|
| Euclidean (L2) | √(Σ(xᵢ − yᵢ)²) | Straight-line distance, geometry, physics |
| Manhattan (L1) | Σ|xᵢ − yᵢ| | Grid navigation, sparse data, feature selection |
| Chebyshev (L∞) | max(|xᵢ − yᵢ|) | Chess king moves, warehouse logistics |
Distance vs. Displacement
Distance is scalar — it measures how far, always positive. Displacement is a vector — it measures how far and in what direction from start to finish. If you walk 3 blocks east then 3 blocks west, your distance is 6 blocks but your displacement is 0. The distance formula calculates displacement magnitude.
Applications
- Navigation and GPS: Haversine formula extends the concept to spherical coordinates for Earth distances.
- Machine learning: K-nearest neighbors, clustering, and similarity metrics all use distance calculations.
- Game development: Collision detection, pathfinding, and range checks.
- Physics: Force fields, orbital mechanics, electromagnetic calculations.
Calculate distance between any two points
Try the Free Distance Calculator →Frequently Asked Questions
What is the distance formula?
d = √((x₂ − x₁)² + (y₂ − y₁)²). It's derived from the Pythagorean theorem — the distance is the hypotenuse of a right triangle formed by the coordinate differences.
How do you calculate distance in 3D?
Add a z-component: d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²). Used in 3D modeling, physics, and aviation.
What is Manhattan distance?
Distance along grid lines: d = |x₂ − x₁| + |y₂ − y₁|. Named after Manhattan's grid layout. Used in logistics, pathfinding, and machine learning.
What is the difference between distance and displacement?
Distance is total path length (always positive). Displacement is straight-line distance with direction (can be zero). The distance formula calculates displacement magnitude.
Can the distance formula give a negative result?
No. Squaring makes differences positive, and the square root is non-negative. The minimum distance is zero (when both points are identical).