Distance Formula Calculator

What Is the Distance Formula?

The distance formula calculates the straight-line (Euclidean) distance between two points in a coordinate system. It is a direct application of the Pythagorean theorem extended into two or three dimensions.

2D Distance Formula

For two points (x₁, y₁) and (x₂, y₂):

d = √((x₂ − x₁)² + (y₂ − y₁)²)

3D Distance Formula

For two points (x₁, y₁, z₁) and (x₂, y₂, z₂):

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

Derivation from the Pythagorean Theorem

In 2D, if you draw a right triangle between two points, the horizontal leg has length |x₂ − x₁|, the vertical leg has length |y₂ − y₁|, and the hypotenuse is the distance d. By Pythagoras: d² = (x₂−x₁)² + (y₂−y₁)² → d = √(…).

The 3D formula is simply the Pythagorean theorem applied twice — first across the horizontal plane, then combined with the vertical difference.

Midpoint Formula

The midpoint between two points is:

Midpoint 2D = ((x₁+x₂)/2, (y₁+y₂)/2)

Midpoint 3D = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)

Slope (2D only)

The slope of the line connecting two points:

m = (y₂ − y₁) / (x₂ − x₁)

A positive slope means the line goes upward from left to right. A negative slope goes downward. A horizontal line has slope 0. A vertical line has undefined slope.

Euclidean vs. Manhattan Distance

Euclidean distance (the straight-line formula above) is the “as the crow flies” distance — the shortest possible path between two points.

Manhattan distance (also called taxicab or city block distance) is the sum of absolute differences:

Manhattan = |x₂ − x₁| + |y₂ − y₁|

Manhattan distance is named after Manhattan’s grid street layout — if you can only travel horizontally or vertically (like blocks in a city), this is the actual travel distance.

Real-World Applications

• GPS navigation: The distance formula calculates straight-line distances between coordinates (latitude/longitude, when simplified)
• Computer graphics: Collision detection, ray casting, determining if a point is inside a circle
• Machine learning: The k-nearest neighbor (KNN) algorithm uses Euclidean distance to find the closest data points to a query
• Game development: Pathfinding algorithms (A* uses distance estimates)
• Robotics: Motion planning, calculating distances between robot joints

Worked Example

Find the distance between points A(2, 3) and B(7, 11): d = √((7−2)² + (11−3)²) = √(25 + 64) = √89 ≈ 9.434

Midpoint: ((2+7)/2, (3+11)/2) = (4.5, 7)

Slope: (11−3)/(7−2) = 8/5 = 1.6