Distance Formula Calculator
Calculate the distance between two points using the Euclidean distance formula
Distance Calculator
Point 1 Coordinates
Point 2 Coordinates
Distance Results
Distance between: (0, 0) and (0, 0)
Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Step-by-Step Solution
Example Calculation
Distance Between Two Cities
Problem: Find the distance between two points on a coordinate plane
Point A: (3, 5)
Point B: (9, 15)
Solution
Step 1: Apply the distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Step 2: Substitute values: d = √[(9 - 3)² + (15 - 5)²]
Step 3: Calculate differences: d = √[6² + 10²]
Step 4: Square the values: d = √[36 + 100]
Step 5: Add: d = √136
Step 6: Final result: d ≈ 11.66 units
Distance Formulas
2D Distance
d = √[(x₂-x₁)² + (y₂-y₁)²]
Euclidean distance between two points
Point to Line
d = |Ax₁+By₁+C|/√(A²+B²)
Distance from point to line
3D Distance
d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
For three-dimensional space
Quick Tips
Distance is always positive (absolute value)
Formula is based on the Pythagorean theorem
Works with negative coordinates
Order of points doesn't matter
Use for coordinate geometry problems
Understanding the Distance Formula
What is 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 gives the shortest distance between any two points.
Applications
- •Coordinate geometry problems
- •Navigation and GPS systems
- •Computer graphics and game development
- •Physics and engineering calculations
Formula Derivation
Pythagorean Theorem: a² + b² = c²
Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
The distance formula comes from creating a right triangle where:
- a = horizontal distance (x₂ - x₁)
- b = vertical distance (y₂ - y₁)
- c = hypotenuse (direct distance)
Remember: The formula gives the straight-line distance, not the distance you would travel along roads or paths.
Types of Distance Calculations
Euclidean Distance
The standard "as the crow flies" distance
- • Most common type
- • Straight-line distance
- • Used in coordinate geometry
- • Basis for other formulas
Manhattan Distance
Distance along grid lines (city blocks)
- • |x₂-x₁| + |y₂-y₁|
- • Used in urban planning
- • Chess rook movement
- • Taxi-cab distance
Chebyshev Distance
Maximum of coordinate differences
- • max(|x₂-x₁|, |y₂-y₁|)
- • Chess king movement
- • Used in game theory
- • Infinity norm