Calculate Distance
Point 1 (x₁, y₁)
Point 2 (x₂, y₂)
Formula & Calculation
Formula
d = √[(x₂-x₁)² + (y₂-y₁)²]
Distance
5.000000
units
Step-by-Step Solution
Visualization
Distance Types Explained
Euclidean Distance
"As the crow flies" - straight-line distance. Uses Pythagorean theorem. Most common for geometric calculations.
Manhattan Distance
Sum of absolute differences in each dimension. Used in grid-based pathfinding, machine learning, and urban navigation.
Haversine Formula
Great-circle distance on a sphere. Accounts for Earth's curvature. Essential for GPS/navigation applications.
3D Distance
Extended Euclidean formula for 3D space. Used in 3D graphics, physics simulations, and spatial analysis.
Distance Result
Distance
5.0000
units
Common Distance Examples
Quick-start with common scenarios
Practice Problems
Test your skills with practice problems
Practice with 3 problems to test your understanding.
?How Do You Calculate Distance?
The distance formula finds the straight-line distance between two points. In 2D: d = sqrt((x2-x1)^2 + (y2-y1)^2). In 3D: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2). This is derived from the Pythagorean theorem. For points (1,2) and (4,6), distance = sqrt(9+16) = sqrt(25) = 5.
What is the Distance Formula?
Distance in coordinate geometry is the length of the straight line connecting two points. The distance formula, derived from the Pythagorean theorem, calculates this Euclidean distance using the differences in coordinates. It is fundamental to geometry, physics, navigation, computer graphics, and many scientific applications.
Key Facts About Distance Calculations
- 2D distance: d = sqrt((x2-x1)^2 + (y2-y1)^2)
- 3D distance: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
- Based on Pythagorean theorem (a^2 + b^2 = c^2)
- Also called Euclidean distance (straight line)
- Manhattan distance uses |x2-x1| + |y2-y1| (grid paths)
- Distance is always positive (absolute value)
- Midpoint formula: ((x1+x2)/2, (y1+y2)/2)
- Used in physics, navigation, graphics, and machine learning
Quick Answer
The distance formula finds the straight-line distance between two points. In 2D: d = sqrt((x2-x1)^2 + (y2-y1)^2). In 3D: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2). This is derived from the Pythagorean theorem. For points (1,2) and (4,6), distance = sqrt(9+16) = sqrt(25) = 5.
Frequently Asked Questions
What is the distance formula?
The distance formula finds the length between two points. In 2D: d = √[(x₂-x₁)² + (y₂-y₁)²]. This is derived from the Pythagorean theorem. In 3D, add (z₂-z₁)² under the square root.
What is Manhattan distance?
Manhattan distance (also called taxicab or L1 distance) measures distance along grid lines, like walking city blocks. It's |x₂-x₁| + |y₂-y₁|. Named after Manhattan's grid street layout where you can't travel diagonally.
How do I calculate distance between GPS coordinates?
Use the Haversine formula for great-circle distance on a sphere. It accounts for Earth's curvature. For short distances (<10km), the simple formula works well. For accuracy, use WGS84 ellipsoid calculations.
When should I use each distance type?
Use 2D Euclidean for flat surfaces and direct "as the crow flies" distance. Use 3D for points in space. Use Manhattan for grid-based movement (game AI, urban navigation). Use Haversine for geographic coordinates on Earth.
What is the difference between distance and displacement?
Distance is the total length traveled along any path (scalar, always positive). Displacement is the shortest straight-line distance between start and end points with direction (vector). Distance ≥ |Displacement|.
Last updated: 2026-01-15
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).