Distance Between Two Points Calculator
Calculate the Euclidean distance between two points in 2D or 3D coordinate space
Calculate Distance Between Points
Point 1 (x₁, y₁)
Point 2 (x₂, y₂)
Distance Calculation Results
Formula Used
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Euclidean distance formula for 2D space
Step by step:
d = √[(0 - 0)² + (0 - 0)²]
d = √[(0)² + (0)²]
d = √[0 + 0]
d = √0
d = 0.000000 units
Coordinate Differences
Δx = x₂ - x₁ = 0
Δy = y₂ - y₁ = 0
Midpoint
(0.000, 0.000)
Slope
Angle
Example Problems
Example 1: Basic 2D Distance
Problem: Find the distance between points (3, 4) and (7, 1).
Solution:
• Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
• d = √[(7 - 3)² + (1 - 4)²]
• d = √[4² + (-3)²] = √[16 + 9] = √25
• Answer: d = 5 units
Example 2: 3D Distance
Problem: Find the distance between points (1, 2, 3) and (4, 6, 8).
Solution:
• Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
• d = √[(4 - 1)² + (6 - 2)² + (8 - 3)²]
• d = √[3² + 4² + 5²] = √[9 + 16 + 25] = √50
• Answer: d = 7.071 units
Example 3: Real-world Application
Problem: A ship is at coordinates (0, 0) and needs to reach a lighthouse at (5, 12) km. What's the direct distance?
Solution:
• d = √[(5 - 0)² + (12 - 0)²]
• d = √[25 + 144] = √169
• Answer: d = 13 km
Distance Formulas
2D Distance
d = √[(x₂-x₁)² + (y₂-y₁)²]
Euclidean distance in plane
3D Distance
d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
Euclidean distance in space
Midpoint
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Center point between coordinates
Key Properties
Shortest Path
Straight line is always shortest distance
Pythagorean Theorem
Based on right triangle principles
Commutative
Distance from A to B equals B to A
Non-negative
Distance is always ≥ 0
Understanding Distance Between Two Points
What is Distance?
In mathematics, distance is a measure of how far apart two points are. The most common type is Euclidean distance, which represents the length of the straight line connecting two points. This is the "as the crow flies" distance that you would measure with a ruler.
Euclidean Distance
Named after the ancient Greek mathematician Euclid, this distance formula is derived from the Pythagorean theorem. It works in any number of dimensions and gives us the shortest possible path between two points in flat space.
Applications
- •Navigation and GPS systems
- •Computer graphics and game development
- •Engineering and architectural planning
- •Physics calculations and scientific research
Mathematical Foundation
Pythagorean Theorem
The distance formula is an extension of the Pythagorean theorem:
c² = a² + b²
Where c is the hypotenuse (distance) and a, b are the legs (coordinate differences).
Coordinate System
Points are defined by their coordinates:
- • 2D: (x, y) - position on a plane
- • 3D: (x, y, z) - position in space
- • Origin: (0, 0) or (0, 0, 0)
Step-by-Step Process
- 1. Find coordinate differences (Δx, Δy, Δz)
- 2. Square each difference
- 3. Add all squared differences
- 4. Take the square root of the sum