Distance Calculator - Calculate 2D, 3D, Geographic & Manhattan Distances with Step-by-Step Solutions

Point 1 (x₁, y₁):

Point 2 (x₂, y₂):

Distance Result

Enter coordinates and click Calculate

Distance Formulas Reference

2D Euclidean Distance

d = √[(x₂-x₁)² + (y₂-y₁)²]

Distance between two points in a plane

Example:

(0,0) to (3,4)
d = √(9 + 16) = 5 units

3D Euclidean Distance

d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]

Distance between two points in space

Example:

(0,0,0) to (1,2,2)
d = √(1 + 4 + 4) = 3 units

Manhattan Distance

d = |x₂-x₁| + |y₂-y₁|

Grid-based distance (taxicab)

Example:

(0,0) to (3,4)
d = 3 + 4 = 7 units

Geographic Distance

Haversine Formula

Great-circle distance on Earth

Example:

NYC to LA
≈ 3,944 km or 2,451 mi

About Distance Calculator

Our Distance Calculator is a comprehensive and free online tool designed to calculate distances between points using various methods including 2D Euclidean distance, 3D spatial distance, geographic distance (using the Haversine formula for GPS coordinates), and Manhattan distance (taxicab geometry). Whether you're a student learning coordinate geometry, a developer working on location-based applications, or a professional needing accurate distance measurements, our calculator provides instant, precise results with detailed step-by-step explanations.

Understanding distance calculations is fundamental to mathematics, physics, geography, computer science, and many real-world applications. Our calculator not only computes distances but also calculates midpoints, shows you the exact formulas used, and provides comprehensive explanations of each calculation method, making it an excellent educational tool as well as a practical utility.

Supported Distance Calculations

  • 2D Euclidean Distance: Calculate straight-line distance between two points on a plane using coordinates (x, y)
  • 3D Spatial Distance: Find distance between two points in three-dimensional space using (x, y, z) coordinates
  • Geographic Distance: Calculate real-world distances using latitude and longitude with the Haversine formula
  • Manhattan Distance: Compute grid-based distance (taxicab geometry) for applications like urban planning and pathfinding
  • Midpoint Calculation: Automatically calculate the midpoint between two points (2D and 3D)
  • Multiple Units: Support for kilometers and miles for geographic distances

Key Features

  • Multiple Distance Types: Four different distance calculation methods in one tool
  • Step-by-Step Solutions: Detailed breakdown of every calculation step
  • Formula Display: See the exact mathematical formula used for each calculation
  • Midpoint Finder: Automatically calculate the center point between two coordinates
  • GPS Support: Enter latitude and longitude coordinates for real-world distances
  • Unit Conversion: Switch between kilometers and miles for geographic distances
  • High Precision: Accurate calculations to 4 decimal places
  • Educational Explanations: Learn why and how each formula works
  • Mobile-Friendly: Works perfectly on all devices
  • 100% Free: No registration or payment required

Understanding Distance Formulas

1. 2D Euclidean Distance

The 2D Euclidean distance formula calculates the straight-line distance between two points in a plane. It is derived from the Pythagorean theorem:

d = √[(x₂-x₁)² + (y₂-y₁)²]

Example: Distance from (0, 0) to (3, 4)
d = √[(3-0)² + (4-0)²] = √[9 + 16] = √25 = 5 units

2. 3D Euclidean Distance

The 3D distance formula extends the 2D formula into three-dimensional space by adding the z-coordinate:

d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]

Example: Distance from (0, 0, 0) to (1, 2, 2)
d = √[(1-0)² + (2-0)² + (2-0)²] = √[1 + 4 + 4] = √9 = 3 units

3. Geographic Distance (Haversine Formula)

The Haversine formula calculates the great-circle distance between two points on a sphere (Earth) given their latitude and longitude. This accounts for Earth's curvature and provides accurate real-world distances:

The formula involves converting degrees to radians and using trigonometric functions to calculate the angular distance, which is then multiplied by Earth's radius (6,371 km or 3,959 miles).

Example: New York City (40.7128°N, 74.0060°W) to Los Angeles (34.0522°N, 118.2437°W)
Distance ≈ 3,944 kilometers or 2,451 miles

4. Manhattan Distance

Manhattan distance (also called taxicab distance or L1 distance) is the sum of the absolute differences of coordinates. It represents the distance traveled on a grid where you can only move horizontally or vertically:

d = |x₂-x₁| + |y₂-y₁|

Example: Distance from (0, 0) to (3, 4)
d = |3-0| + |4-0| = 3 + 4 = 7 units
(Compare to Euclidean: 5 units)

Midpoint Formula

The midpoint is the point exactly halfway between two given points. It is calculated by averaging the coordinates:

2D: M = ((x₁+x₂)/2, (y₁+y₂)/2)
3D: M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)

Example: Midpoint between (0, 0) and (6, 8)
M = ((0+6)/2, (0+8)/2) = (3, 4)

Practical Applications

2D & 3D Distance Applications

  • Computer Graphics: Calculate distances between objects, collision detection, rendering optimization
  • Game Development: Character movement, AI pathfinding, proximity detection
  • Robotics: Path planning, obstacle avoidance, sensor measurements
  • Physics & Engineering: Calculate displacement, velocity, and force vectors
  • Machine Learning: K-nearest neighbors (KNN), clustering algorithms, similarity measures
  • Architecture: Spatial planning, structural analysis, 3D modeling

Geographic Distance Applications

  • Navigation & GPS: Route planning, travel distance estimation, location-based services
  • Logistics & Delivery: Optimize delivery routes, calculate shipping costs, fleet management
  • Travel Planning: Estimate travel distances, compare flight paths, plan road trips
  • Real Estate: Calculate proximity to amenities, compare locations, property valuation
  • Emergency Services: Find nearest hospitals, fire stations, optimize response times
  • Wildlife Tracking: Monitor animal movements, migration patterns, habitat analysis

Manhattan Distance Applications

  • Urban Planning: Calculate city block distances, analyze street networks
  • Warehouse Management: Optimize picking routes, inventory placement
  • Circuit Board Design: Calculate wire lengths, routing optimization
  • Chess & Board Games: Calculate move distances for kings and rooks
  • Image Processing: Pixel distance calculations, edge detection
  • Data Science: Feature scaling, outlier detection, classification algorithms

How to Use the Distance Calculator

For 2D Distance:

  1. Select the "2D Distance" tab
  2. Enter the x and y coordinates for Point 1
  3. Enter the x and y coordinates for Point 2
  4. Click "Calculate Distance"
  5. View the distance, midpoint, and step-by-step solution

For 3D Distance:

  1. Select the "3D Distance" tab
  2. Enter the x, y, and z coordinates for Point 1
  3. Enter the x, y, and z coordinates for Point 2
  4. Click "Calculate Distance"
  5. View the 3D distance, midpoint, and calculation steps

For Geographic Distance:

  1. Select the "Geographic" tab
  2. Choose your preferred unit (Kilometers or Miles)
  3. Enter latitude and longitude for Location 1 (e.g., 40.7128, -74.0060 for New York)
  4. Enter latitude and longitude for Location 2 (e.g., 34.0522, -118.2437 for Los Angeles)
  5. Click "Calculate Distance"
  6. View the great-circle distance and detailed calculation using the Haversine formula

For Manhattan Distance:

  1. Select the "Manhattan" tab
  2. Enter the x and y coordinates for Point 1
  3. Enter the x and y coordinates for Point 2
  4. Click "Calculate Distance"
  5. View the Manhattan distance and see how it compares to Euclidean distance

Distance Calculation Examples

Example 1: 2D Euclidean Distance

Problem: Find the distance between points A(1, 2) and B(4, 6)

Solution:

d = √[(4-1)² + (6-2)²]
d = √[3² + 4²]
d = √[9 + 16]
d = √25
d = 5 units

Midpoint: M = ((1+4)/2, (2+6)/2) = (2.5, 4)

Example 2: 3D Distance

Problem: Calculate distance between points P(1, 2, 3) and Q(4, 6, 8)

Solution:

d = √[(4-1)² + (6-2)² + (8-3)²]
d = √[3² + 4² + 5²]
d = √[9 + 16 + 25]
d = √50
d ≈ 7.0711 units

Example 3: Geographic Distance

Problem: Distance from London (51.5074°N, 0.1278°W) to Paris (48.8566°N, 2.3522°E)

Solution: Using Haversine formula

Distance ≈ 344 kilometers or 214 miles

Example 4: Manhattan Distance

Problem: Manhattan distance from (0, 0) to (5, 7)

Solution:

d = |5-0| + |7-0|
d = 5 + 7
d = 12 units

Note: Euclidean distance would be approximately 8.6 units

Tips for Distance Calculations

  1. Check Your Units: Ensure all coordinates use the same units (meters, feet, etc.)
  2. Geographic Coordinates: Remember that latitude ranges from -90° to 90° and longitude from -180° to 180°
  3. Negative Coordinates: Distance formulas work with negative numbers; the result is always positive
  4. Choose the Right Method: Use Euclidean for straight-line, Manhattan for grid-based, Geographic for Earth surface
  5. Verify Results: Check if your calculated distance makes sense given the input coordinates
  6. Precision Matters: For GPS coordinates, more decimal places = higher precision (6 decimals ≈ 0.1 meter accuracy)
  7. Earth's Shape: For very long distances, consider that Earth is not perfectly spherical (use WGS84 ellipsoid for highest precision)

Common Distance Values Reference

Pythagorean Triples (2D):

Common right triangles where all sides are integers:

  • • 3-4-5 triangle: Distance from (0,0) to (3,4) = 5 units
  • • 5-12-13 triangle: Distance from (0,0) to (5,12) = 13 units
  • • 8-15-17 triangle: Distance from (0,0) to (8,15) = 17 units
  • • 7-24-25 triangle: Distance from (0,0) to (7,24) = 25 units

Major City Distances:

  • • New York to Los Angeles: ~3,944 km / 2,451 mi
  • • London to Paris: ~344 km / 214 mi
  • • Tokyo to Beijing: ~2,100 km / 1,305 mi
  • • Sydney to Melbourne: ~713 km / 443 mi

Why Use Our Distance Calculator?

Our Distance Calculator stands out for its versatility, accuracy, and educational value. Unlike basic calculators that only show final results, we provide comprehensive step-by-step solutions that help you understand the mathematical process behind each calculation. The support for multiple distance types (2D, 3D, geographic, and Manhattan) makes it a one-stop solution for all your distance calculation needs.

Whether you need to calculate the distance between two points on a graph, find the real-world distance between cities using GPS coordinates, or determine grid-based distances for pathfinding algorithms, our calculator handles it all with ease. The automatic midpoint calculation feature and detailed formula explanations make it an excellent tool for both learning and practical applications.

The intuitive interface works perfectly on desktop and mobile devices, ensuring you can perform distance calculations wherever you are. With support for multiple coordinate systems, unit conversions, and high-precision calculations, our calculator is trusted by students, educators, developers, and professionals worldwide. Best of all, it's completely free with no registration required.