Distance Calculator
2D Distance Calculator
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Example:
For points (1,1) and (4,5):
√[(4-1)² + (5-1)²] = √(9 + 16) = 5
3D Distance Calculator
Distance = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
Example:
For points (1,1,1) and (2,2,2):
√[(2-1)² + (2-1)² + (2-1)²] = √(1 + 1 + 1) ≈ 1.732
Distance Based on Latitude and Longitude
a = sin²(Δφ/2) + cos(φ₁) × cos(φ₂) × sin²(Δλ/2)
c = 2 × atan2(√a, √(1−a))
Distance = R × c (where R is Earth's radius: 6371 km)
Example (Decimal Degrees):
Point 1: (38.8976° N, 77.0366° W)
Point 2: (39.9496° N, 75.1503° W)
Δφ = 1.0520°, Δλ = 1.8863°
Distance ≈ 199.3 km (123.9 miles)
Example (DMS):
Point 1: 38°53′51″ N, 77°2′12″ W
Point 2: 39°56′59″ N, 75°9′1″ W
Converted to decimal:
(38.8975°, 77.0367°) and (39.9497°, 75.1503°)
Δφ = 1.0522°, Δλ = 1.8864°
Distance ≈ 199.3 km (123.9 miles)
Distance in a 2D Coordinate Plane
To calculate the distance between two points on a flat surface (2D plane), you can use the Distance Formula, derived from the Pythagorean Theorem.
Formula:
Distance = √[ (x₂ – x₁)² + (y₂ – y₁)² ]
Here,
(x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
The order of points doesn’t matter, as squaring removes the sign.
Example:
Find the distance between (1, 5) and (3, 2):
You can switch the point order, and the result will still be the same.
Distance in a 3D Coordinate Space
When working with three-dimensional coordinates, simply extend the 2D distance formula to include the z-axis.
Formula:
Example:
Given points (1, 3, 7) and (2, 4, 8):
This formula is ideal for calculating the straight-line distance in 3D modeling, physics, and simulations.
Distance on Earth’s Surface
Measuring distance on the Earth requires more advanced calculations due to its spherical (actually ellipsoidal) shape. Two commonly used formulas are:
Haversine Formula (for Spherical Earth Approximation)
The Haversine Formula is used to calculate the shortest path between two points on a sphere (like Earth, approximated).
Formula:
Note:
This formula assumes the Earth is a perfect sphere. It’s widely used in GPS applications but can be off by ~0.5% due to Earth’s actual ellipsoidal shape.
Lambert’s Formula (for Ellipsoidal Earth Surface)
Lambert’s Formula provides more accurate results by considering Earth as an ellipsoid rather than a perfect sphere.
Used for:
High-accuracy distance measurement in mapping systems, GIS, and geolocation services.
Formula (Simplified Form):
Where:
a = Earth’s equatorial radius
σ = central angle between two points in radians
f = Earth’s flattening factor
β = reduced latitude:
Lambert’s method achieves high precision (within ~10 meters) across long distances—far superior to the Haversine approximation.
Quick Comparison Table
| Type of Distance | Formula Used |
|---|---|
| 2D Plane | √[ (x₂ – x₁)² + (y₂ – y₁)² ] |
| 3D Space | √[ (x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)² ] |
| Earth’s Surface (Spherical) | Haversine Formula |
| Earth’s Surface (Ellipsoidal) | Lambert’s Formula |
Applications
Geometry and math problems
3D simulations and graphics
Route mapping and travel planning
GPS and navigation tools
Surveying and land measurements