Distance Between Two Points Calculator
Calculate distance and midpoint between two coordinate points
Table of Contents
How to Use
- Enter the x-coordinate of the first point (x₁)
- Enter the y-coordinate of the first point (y₁)
- Enter the x-coordinate of the second point (x₂)
- Enter the y-coordinate of the second point (y₂)
- Click Calculate to see the distance, midpoint, and component distances
The Distance Formula
The distance formula calculates the straight-line distance between two points in a coordinate plane. It's derived from the Pythagorean theorem and is one of the most fundamental formulas in coordinate geometry.
For two points (x₁, y₁) and (x₂, y₂), the distance d is: d = √[(x₂-x₁)² + (y₂-y₁)²]
Derivation from Pythagorean Theorem
The distance formula comes from the Pythagorean theorem. If you draw a right triangle with the two points as opposite corners:
- The horizontal leg has length |x₂ - x₁|
- The vertical leg has length |y₂ - y₁|
- The hypotenuse is the distance between the points
- By Pythagorean theorem: d² = (x₂-x₁)² + (y₂-y₁)²
- Taking the square root gives the distance formula
Midpoint Formula
The midpoint is the point exactly halfway between two points. It's calculated by averaging the x-coordinates and y-coordinates separately.
Midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2)
The midpoint divides the line segment connecting the two points into two equal parts.
Special Cases
| Case | Condition | Result |
|---|---|---|
| Same point | (x₁, y₁) = (x₂, y₂) | Distance = 0 |
| Horizontal line | y₁ = y₂ | Distance = |x₂ - x₁| |
| Vertical line | x₁ = x₂ | Distance = |y₂ - y₁| |
| Origin to point | (x₁, y₁) = (0, 0) | Distance = √(x₂² + y₂²) |
Real-World Applications
- Navigation: GPS systems calculate distances between coordinates
- Computer graphics: Rendering and collision detection
- Robotics: Path planning and obstacle avoidance
- Game development: Character movement and AI
- Data science: Clustering algorithms (k-means, etc.)
- Physics: Calculating displacement and velocity
- Architecture: Measuring distances on blueprints
- Astronomy: Calculating distances between celestial objects
Frequently Asked Questions
- What's the difference between distance and displacement?
- Distance is the total length of the path traveled, while displacement is the straight-line distance from start to finish. The distance formula calculates displacement (the shortest path between two points).
- Can the distance formula be used in 3D space?
- Yes! The 3D distance formula is d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]. It extends the 2D formula by adding the z-component.
- Why do we square the differences before adding them?
- Squaring ensures all values are positive (eliminating direction) and comes from the Pythagorean theorem. It's the mathematical way to combine perpendicular components into a total distance.
- Is the distance formula the same as the Euclidean distance?
- Yes, the distance formula calculates Euclidean distance, which is the 'ordinary' straight-line distance in Euclidean geometry. There are other distance metrics (Manhattan, Chebyshev) used in specific applications.