What is the distance formula?

The distance formula calculates the straight-line (Euclidean) distance between two points in a coordinate system. In 2D: d = √((x₂−x₁)² + (y₂−y₁)²). In 3D: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). The formula is derived directly from the Pythagorean theorem — the horizontal difference (Δx) and vertical difference (Δy) form the legs of a right triangle, and the distance is the hypotenuse.

How do you find the distance between two points step by step?

To find the distance between (x₁, y₁) and (x₂, y₂): (1) Calculate Δx = x₂ − x₁. (2) Calculate Δy = y₂ − y₁. (3) Square both: (Δx)² and (Δy)². (4) Add them: (Δx)² + (Δy)². (5) Take the square root. For points (1, 2) and (4, 6): Δx = 3, Δy = 4, (Δx)² + (Δy)² = 9 + 16 = 25, d = √25 = 5. The order of the points does not matter — distance is always positive.

What is Euclidean distance?

Euclidean distance is the "ordinary" straight-line distance between two points in Euclidean space — the distance you would measure with a ruler. It is named after the ancient Greek mathematician Euclid. The Euclidean distance is one of several distance metrics: others include Manhattan distance (sum of absolute coordinate differences), Chebyshev distance (maximum coordinate difference), and Minkowski distance (a generalization). For everyday geometry, Euclidean distance is the standard. In machine learning and data science, different distance metrics are chosen depending on the structure of the data.

How do you find distance in 3D?

The 3D distance formula extends the 2D version by adding the z-component: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). For points A = (1, 2, 3) and B = (4, 6, 8): Δx = 3, Δy = 4, Δz = 5, d = √(9 + 16 + 25) = √50 ≈ 7.07. This is equivalent to applying the Pythagorean theorem twice: first in the horizontal plane to get the horizontal distance, then in the vertical plane using that horizontal distance and the height difference.

How is the distance formula related to the Pythagorean theorem?

The distance formula is the Pythagorean theorem applied to coordinate geometry. For two points (x₁, y₁) and (x₂, y₂), the horizontal difference Δx and vertical difference Δy are the two legs of a right triangle, and the straight-line distance d is the hypotenuse: d² = (Δx)² + (Δy)², so d = √((Δx)² + (Δy)²). This is exactly a² + b² = c² from the Pythagorean theorem. Our Pythagorean theorem calculator solves for any side of a right triangle given the other two.

What is the difference between distance and displacement?

Distance is a scalar — it measures the total length of a path traveled, always positive, regardless of direction. Displacement is a vector — it measures the straight-line change in position from start to finish, including direction. If you walk 4 miles east and then 3 miles north, your total distance traveled is 7 miles, but your displacement is 5 miles (northeast) — the straight-line distance between start and end points. The distance formula calculates displacement (straight-line distance), not total path length.

How is the distance formula used in machine learning?

In machine learning, Euclidean distance is the foundation of many algorithms. k-Nearest Neighbors (k-NN) classifies a data point by finding the k training examples with the smallest Euclidean distance and taking a majority vote. k-Means clustering assigns each data point to the cluster whose centroid is closest (by Euclidean distance). Support Vector Machines maximize the Euclidean distance between the decision boundary and the nearest data points (the "margin"). In recommendation systems, items or users are represented as vectors and similar ones are identified by their small Euclidean distances.

What is the Manhattan distance and how is it different from Euclidean distance?

Manhattan distance (also called taxicab distance or L1 distance) is the sum of the absolute differences of coordinates: d = |x₂−x₁| + |y₂−y₁|. It measures the distance you would travel on a grid where you can only move horizontally or vertically — like a taxi navigating city blocks. For points (1,1) and (4,5): Euclidean distance = √(9+16) = 5; Manhattan distance = 3 + 4 = 7. Manhattan distance is preferred in high-dimensional data where the "curse of dimensionality" makes Euclidean distance less meaningful, and in applications involving grid-based movement (robotics, logistics).

Distance Calculator — Between Two Points (2D & 3D)

Calculates the Euclidean distance between two points in 2D or 3D, plus midpoint and slope.

How to Use the Distance Calculator

This distance calculator finds the straight-line Euclidean distance between any two points in 2D or 3D space — select a mode, enter your coordinates, and it instantly returns the distance, the differences Δx and Δy (and Δz in 3D), the midpoint, and — in 2D mode — the slope. All inputs accept any real number, including negatives and decimals. For finding the center of a segment, see our midpoint calculator.

Use the Share button to copy a shareable link with your inputs pre-filled — useful for homework verification, collaboration, or saving calculations.

The Distance Formula

The distance formula computes the straight-line Euclidean distance between two points:

2D: d = √((x₂ − x₁)² + (y₂ − y₁)²)

3D: d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)

This is the Pythagorean theorem in coordinate form. For points (3, 0) and (0, 4): d = √((0−3)² + (4−0)²) = √(9 + 16) = √25 = 5. This is the classic 3-4-5 right triangle.

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Distance Formula vs. Pythagorean Theorem

The distance formula and the Pythagorean theorem are the same relationship expressed in different contexts. In both cases, two legs form a right angle and the hypotenuse (or distance) satisfies:

d² = (Δx)² + (Δy)²

The Pythagorean theorem is typically used when you are working with a right triangle directly (given side lengths). The distance formula is used when you have coordinate points and want to find the length of the segment between them. They are two ways of expressing the same geometric truth. Our Pythagorean theorem calculator lets you solve for any missing side given the other two.

Manhattan Distance vs. Euclidean Distance

Not all distance problems call for Euclidean (straight-line) distance. Here are the most common distance metrics and when each applies:

Euclidean distance — straight-line distance, as the crow flies. Used in geometry, physics, GPS, and most real-world distance problems.
Manhattan distance (taxicab distance) — |Δx| + |Δy|. The distance traveled along a grid, like city blocks. Used when movement is constrained to horizontal/vertical paths.
Chebyshev distance — max(|Δx|, |Δy|). How many king moves in chess to travel from one square to another. Used in board games and some robotics problems.
Great-circle distance — distance along the surface of a sphere, using latitude/longitude. Used for real-world navigation and geographic distance between locations.

Real-World Applications of the Distance Formula

The distance formula is used across many fields:

Computer graphics — calculating distances between objects for collision detection, rendering, and path finding
GPS and navigation — computing straight-line distances between two geographic coordinates (simplified as Euclidean for short distances)
Machine learning — k-nearest neighbors (kNN) classification uses Euclidean distance to find the closest training examples
Physics — displacement between two positions in space
Engineering — verifying distances in CAD drawings and structural models

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Step-by-Step Example: 3D Distance

Problem: Find the distance between P₁ = (1, 2, 3) and P₂ = (4, 6, 15).

Δx = 4 − 1 = 3
Δy = 6 − 2 = 4
Δz = 15 − 3 = 12
(Δx)² + (Δy)² + (Δz)² = 9 + 16 + 144 = 169
d = √169 = 13

This is a 3D Pythagorean triple: 3² + 4² + 12² = 9 + 16 + 144 = 169 = 13². Midpoint: ((1+4)/2, (2+6)/2, (3+15)/2) = (2.50, 4.00, 9.00).

Sources & References

Distance Formula — Khan Academy
Euclidean Distance — Math Is Fun

Distance Calculator