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

0.000000
Distance (units)
2D
Coordinate System

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

Undefined

Angle

0.000°

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. 1. Find coordinate differences (Δx, Δy, Δz)
  2. 2. Square each difference
  3. 3. Add all squared differences
  4. 4. Take the square root of the sum