Distance Calculator – Calculate 2D, 3D & Geographic Distance with Haversine Formula

  • Adam
  • November 4, 2025

Free online distance calculator for 2D coordinates, 3D space, and geographic locations (latitude/longitude). Uses Haversine formula for accurate earth distance. Step-by-step solutions.

Distance Calculator

Calculate Distance in 2D Plane, 3D Space, and Geographic Coordinates with Complete Analysis

Quick Navigation

2D Distance Calculator

Calculate distance between two points on a 2D coordinate plane.

2D Distance Calculator
Point 1 (x₁, y₁)
Point 2 (x₂, y₂)
Distance (d)
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Δx
-
Δy
-
Visual Reference
(x₁,y₁) (x₂,y₂) Δx Δy d
Formula:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]

3D Distance Calculator

Calculate distance between two points in 3D coordinate space.

3D Distance Calculator
Point 1 (x₁, y₁, z₁)
Point 2 (x₂, y₂, z₂)
Distance (d)
-
Δx
-
Δy
-
Δz
-
Visual Reference
x z y P₁ P₂
Formula:
d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]

Geographic Distance Calculator (Haversine Formula)

Calculate distance between two points on Earth using latitude and longitude coordinates.

Geographic Distance
Location 1
Location 2
Distance
-
Earth Reference
🌍
Haversine Formula
Method: Great-circle distance
Accuracy: ±0.5 km on Earth's surface
Formula: Based on spherical geometry

Understanding Distance

Distance measures the separation between two points in space. The most direct distance is the straight line connecting them. Distance calculations are fundamental across mathematics, physics, geography, navigation, and countless practical applications from surveying to video game development.

Three main types of distance calculations serve different purposes: 2D distance works on flat planes using Euclidean geometry; 3D distance extends this to three-dimensional space; geographic distance accounts for Earth's curvature using spherical geometry. Each requires its own formula based on the coordinate system and dimensional space involved.

The distance formula is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. This principle extends beautifully to three dimensions and beyond.

Key Principle: Distance is always the shortest path between two points (straight line in Euclidean space, great circle on sphere).

Key Features & Capabilities

2D Coordinates Calculate distance on flat planes with x, y coordinates
3D Coordinates Find distance in 3D space with x, y, z coordinates
Geographic Coordinates Use latitude and longitude for Earth surface distances
Haversine Formula Accounts for Earth's spherical shape accurately
Multiple Units Display results in km, miles, nautical miles, meters
Step-by-Step Detailed calculation breakdown shown for each distance

How to Use This Calculator

For 2D Distance

  1. Identify Coordinates: Know both (x₁, y₁) and (x₂, y₂)
  2. Enter Point 1: Input x₁ and y₁ values
  3. Enter Point 2: Input x₂ and y₂ values
  4. Calculate: Press Calculate button
  5. Review: See distance and component distances (Δx, Δy)

For 3D Distance

  1. Gather Coordinates: Know (x₁, y₁, z₁) and (x₂, y₂, z₂)
  2. Enter Point 1: Input all three coordinates
  3. Enter Point 2: Input all three coordinates
  4. Calculate: Press Calculate button
  5. Review: See 3D distance and all components

For Geographic Distance

  1. Gather Coordinates: Know latitude/longitude for both points
  2. Choose Unit: Select km, miles, nautical miles, or meters
  3. Enter Coordinates: Input lat/lon for both locations
  4. Calculate: Press Calculate button
  5. Get Distance: See great-circle distance between locations

Tips for Accurate Calculations

  • Coordinate Order: Be consistent with point labels (Point 1 vs Point 2)
  • Units Consistency: All coordinates must use same unit system
  • Geographic Precision: Use decimal degrees for latitude/longitude (e.g., 38.8976, not 38°53'52")
  • Latitude Range: Must be between -90° and +90°
  • Longitude Range: Must be between -180° and +180°

Complete Formulas Reference

2D Distance Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where (x₁, y₁) and (x₂, y₂) are the coordinates of two points
3D Distance Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]

Extension of 2D formula adding the z-coordinate component
Haversine Formula (Geographic)
a = sin²(Δφ/2) + cos(φ₁) × cos(φ₂) × sin²(Δλ/2)
c = 2 × atan2(√a, √(1−a))
d = R × c

Where φ = latitude, λ = longitude, R = Earth radius (6,371 km)

Core Concepts Explained

Pythagorean Theorem Foundation

The 2D distance formula is based on the Pythagorean theorem: \(a^2 + b^2 = c^2\). If we create a right triangle where horizontal distance is one leg and vertical distance is the other leg, the hypotenuse is the direct distance between points.

Euclidean vs Spherical Distance

Euclidean distance applies to flat planes and 3D space. Spherical distance (great-circle distance) applies to Earth's surface. The difference becomes significant over long distances. A 1,000 km journey follows a curve (great circle) rather than straight line on Earth's spherical surface.

Components of Distance

Distance has components: horizontal change (Δx), vertical change (Δy), and in 3D, depth change (Δz). The total distance combines all components using the Pythagorean principle. Understanding components helps visualize the geometry.

Great-Circle Distance

The shortest distance between two points on a sphere is along the great circle (circle with same center as sphere). Flight paths follow great circles. The Haversine formula calculates this accurately, accounting for Earth's shape.

Real-World Applications

Navigation and Travel

GPS systems use geographic distance calculations to determine travel distances between locations. Flight paths follow great-circle routes (shortest on Earth's surface). Driving directions use 3D distance principles adjusted for roads.

Architecture and Engineering

Architects use 2D distance for floor plans, 3D for spatial relationships. Structural engineers calculate distances for load paths and stress distribution. Surveyors use geographic distance to map land areas.

Physics and Science

Physicists calculate distances between particles and celestial objects. 3D distance essential for understanding molecular structure and atomic interactions. Distance squared appears in gravitational and electromagnetic force equations.

Video Games and Graphics

Game engines constantly calculate distances for collision detection, visibility culling, and AI behavior. 3D distance determines when objects are close enough to interact. Rendering systems use distance for level-of-detail decisions.

Data Science and Analytics

Machine learning uses distance metrics for clustering and classification. K-nearest neighbors algorithm relies on distance calculations. Pattern recognition depends on measuring distances in multidimensional data space.

Worked Examples

Example 1: 2D Distance - Map Coordinates

Problem: Two cities at coordinates (1, 5) and (4, 9). Find distance.

Solution:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
d = √[(4 - 1)² + (9 - 5)²]
d = √[3² + 4²]
d = √[9 + 16]
d = √25 = 5 units

Example 2: 3D Distance - Building Coordinates

Problem: Two points (1, 3, 7) and (2, 4, 8). Find distance.

Solution:
d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
d = √[(2 - 1)² + (4 - 3)² + (8 - 7)²]
d = √[1² + 1² + 1²]
d = √[1 + 1 + 1]
d = √3 ≈ 1.732 units

Example 3: Geographic Distance - Cities

Problem: Washington DC (38.8976°N, -77.0366°W) to New York (40.7128°N, -74.0060°W). Find distance.

Solution:
Using Haversine formula:
Δφ = 40.7128 - 38.8976 = 1.8152°
Δλ = -74.0060 - (-77.0366) = 3.0306°
Calculation yields:
Distance ≈ 360 km (≈225 miles) great-circle distance

Example 4: Right Triangle 2D

Problem: Points (0, 0) and (3, 4) forming 3-4-5 triangle. Find distance.

Solution:
d = √[(3 - 0)² + (4 - 0)²]
d = √[9 + 16]
d = √25 = 5
Classic 3-4-5 Pythagorean triple

Frequently Asked Questions

❓ Why use different formulas for different dimensions?
Because coordinates have different dimensions. 2D uses x and y. 3D adds z. Geographic uses latitude/longitude on a sphere, not a flat plane. Each system requires appropriate geometry.
❓ Is distance always positive?
Yes. Distance is the absolute separation, always positive. Even if coordinates are negative, the squared terms make result positive, and the square root gives positive distance.
❓ What's the difference between distance and displacement?
Distance is total path length traveled (always positive). Displacement is straight-line separation from start to end (can be negative in 1D). This calculator computes displacement/straight-line distance.
❓ Why is Haversine better than simple geographic distance?
Haversine accounts for Earth's spherical shape. Simple formulas treat Earth as flat, causing significant errors over long distances. Haversine accurate to within 0.5 km on Earth's surface.
❓ Can I use this for distances on other planets?
Yes, for 2D/3D distance. For geographic (Haversine), you'd need to change Earth's radius to the planet's radius. Mars radius is 3,389 km instead of 6,371 km.
❓ What's the Manhattan distance?
Manhattan distance (taxicab distance) is |x₂ - x₁| + |y₂ - y₁|. It measures distance along grid lines, not straight line. This calculator uses Euclidean distance (straight line).
❓ How accurate is the geographic distance?
Haversine formula accurate to ±0.5 km on Earth. More precise Lambert's formula available for extreme accuracy. For practical purposes, Haversine is excellent.

Calculate Distance Instantly

Whether you're planning navigation, analyzing spatial data, designing structures, or solving geometry problems, this comprehensive distance calculator handles 2D, 3D, and geographic coordinates with instant accuracy. Fast, reliable, completely free.

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