# Distance Between Points Calculator

Calculate the distance between two points in 2D or 3D space. Shows Euclidean distance, Manhattan distance, midpoint, and step-by-step work. Free calculator.

## What this calculates

Calculate the straight-line distance between any two points in 2D or 3D space. This calculator also provides the Manhattan distance, the midpoint, and a step-by-step breakdown of the formula.

## Inputs

- **Dimensions** — options: 2D (x, y), 3D (x, y, z) — Choose 2D or 3D coordinate space.
- **Point 1: X**
- **Point 1: Y**
- **Point 1: Z** — Only used in 3D mode.
- **Point 2: X**
- **Point 2: Y**
- **Point 2: Z** — Only used in 3D mode.

## Outputs

- **Distance** — The straight-line (Euclidean) distance between the two points.
- **Distance Squared** — The square of the distance.
- **Manhattan Distance** — Sum of absolute differences along each axis.
- **Midpoint** — formatted as text — The point exactly halfway between the two points.
- **Calculation Steps** — formatted as text — Step-by-step breakdown using the distance formula.

## Details

The Euclidean distance formula measures the straight-line distance between two points. It is a generalization of the Pythagorean theorem to any number of dimensions.

**2D Distance Formula:**

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

**3D Distance Formula:**

d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

**Manhattan Distance:**

Also called taxicab or L1 distance, it sums the absolute differences along each axis: |x2-x1| + |y2-y1| (+ |z2-z1| in 3D). It measures the distance you would travel on a grid, like city blocks.

**Midpoint Formula:**

The midpoint is the average of the coordinates: ((x1+x2)/2, (y1+y2)/2) in 2D. It is the point exactly halfway between the two given points.

**Applications:**

Distance calculations are essential in GPS navigation, game development (collision detection), machine learning (k-nearest neighbors), physics (force calculations), and everyday tasks like finding the shortest route between locations.

## Frequently Asked Questions

**Q: How do I find the distance between two points in 3D?**

A: Use the 3D distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2). It works the same as the 2D formula but adds the z-coordinate difference. For points (1, 2, 3) and (4, 6, 8): d = sqrt(9 + 16 + 25) = sqrt(50) = 7.071.

**Q: What is the difference between Euclidean and Manhattan distance?**

A: Euclidean distance is the straight-line (as-the-crow-flies) distance between two points. Manhattan distance is the sum of absolute differences along each axis, like walking along a grid of city blocks. Euclidean is always less than or equal to Manhattan distance. They are equal only when the points differ along just one axis.

**Q: Does the order of the points matter?**

A: No. Distance from point A to point B is always the same as from B to A. The formula squares the differences, so the sign does not matter. This is true for both Euclidean and Manhattan distance.

**Q: What is the midpoint used for?**

A: The midpoint is the point exactly halfway between two points. It is useful for finding the center of a line segment, bisecting shapes, locating the center of mass between two equal masses, and as a starting point for binary search algorithms in geometry.

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Source: https://vastcalc.com/calculators/math/distance-between-two-points
Category: Math
Last updated: 2026-04-08