Distance Calculator

2D Distance Calculator (Plane Coordinates)

Point 1 (x1, y1)

Point 2 (x2, y2)

2D Distance Result

3D Distance Calculator (3D Space Coordinates)

Point 1 (x1, y1, z1)

Point 2 (x2, y2, z2)

3D Distance Result

Distance Based on Latitude and Longitude

Point 1 (Latitude 1 in degrees, Longitude 1 in degrees)

Point 2 (Latitude 2 in degrees, Longitude 2 in degrees)

Latitude/Longitude Distance Result

Great Circle Distance Calculator (Interactive Map)

Click on the map to place two markers, then drag them to adjust positions. The shortest great circle distance will be calculated automatically.

Great Circle Distance Result

How to Use This Distance Calculator: A Step-by-Step Guide

Our free online Distance Calculator handles Euclidean distances in 2D/3D planes and great circle distances on Earth's surface, with an interactive map for visual placement. Input coordinates or click markers for instant results in units like km or miles, using precise formulas like Haversine. Client-side computation ensures privacy and speed—perfect for students, hikers, developers, or pilots. Responsive across devices, with error handling for invalid inputs.

Choose Calculator Type: Scroll to 2D (plane points), 3D (space vectors), Lat/Lon (geographic coords), or Map (visual). Enter x,y (or z) values; for geo, use degrees (lat -90° to 90°, lon -180° to 180°).
Input Coordinates: Fill fields (e.g., 2D: (1,2) and (4,6)). For map, click twice to place draggable markers—distance updates live with a dashed route.
Calculate: Hit Calculate; results show below with rounded precision (2 decimals default). Geo outputs km and miles; clear via buttons to reset.
Validate & Troubleshoot: Invalid numbers (NaN, out-of-range) flag errors with messages. Zoom/pan map for global views; drag markers for refinements.
Explore Outputs: Results include distance values; use for path planning or simulations. Keyboard: Tab/Enter for inputs; mobile: touch-drag markers.
Learn More Below: Dive into derivations, examples, tables, apps, and tips for each method, from Pythagoras to spherical trig in 2025's GIS and VR worlds.

Pro Tip: For aviation, use great circle—shortest paths curve! Test with NYC (40.7128°N, -74.0060°W) to London (51.5074°N, 0.1278°W) ≈5570 km. Bookmark for fieldwork.

Distance Formulas and Examples

2D Euclidean Distance: Formula, Derivation, and Detailed Examples

The 2D Euclidean distance quantifies straight-line separation in Cartesian planes, rooted in Pythagoras' theorem (c. 500 BC). Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]. Derivation: Form a right triangle with legs Δx and Δy; hypotenuse d via vector dot product |v₂ - v₁|. Essential in GIS for flat projections and robotics pathfinding.

In 2025, it's core to AR overlays aligning virtual objects in real 2D spaces.

Step-by-Step Example: (1,2) to (4,6):
1. Δx = 4-1=3, Δy=6-2=4.
2. (3)² + (4)² = 9+16=25.
3. √25=5 units. Tool computes instantly.

Point 1 (x,y)	Point 2 (x,y)	Distance
(0,0)	(3,4)	5
(1,2)	(4,6)	5
(-1,1)	(2,-3)	5.385

Real-World Applications: Game sprites movement (e.g., 3Δx+4Δy=5 pixels). Urban planning block distances.

Tip: Pre-square deltas to avoid negatives; verify with Manhattan for grids.
Common Error: Forgetting squares—use (Δx)² not Δx.
Advanced: Minkowski p=2 norm generalizes to other metrics.

Example: Distance Between (1,2) and (4,6)

Using points (1,2) and (4,6), the distance is 5 units. This could represent the distance a robot travels on a flat factory floor between two waypoints, avoiding obstacles by following the shortest path.

3D Euclidean Distance: Formula, Derivation, and Detailed Examples

Extends 2D to volume with z-delta. Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]. Derivation: Pythagoras in 3D via vector magnitude ||v|| = √(v·v); from Apollonius' conics to modern tensor norms.

3D distances model molecular bonds. In 2025, vital for volumetric rendering in metaverse navigation.

Step-by-Step Example: (0,0,0) to (3,4,5):
1. Δx=3, Δy=4, Δz=5.
2. 9+16+25=50.
3. √50≈7.07 units.

Point 1 (x,y,z)	Point 2 (x,y,z)	Distance
(0,0,0)	(1,1,1)	1.73
(0,0,0)	(3,4,5)	7.07
(1,2,3)	(4,6,8)	7.07

Real-World Applications: Drone flight vectors (3D coords=7.07m). Protein folding simulations.

Tip: Sum squares first; use hypot() in code for overflow safety.
Common Error: Unit mismatches (m vs cm)—standardize.
Advanced: Mahalanobis for correlated dimensions in ML.

Example: Distance Between (0,0,0) and (3,4,5)

Calculate the distance as \(\sqrt{50} \approx 7.07\) units. In architecture, this measures the direct path between two points in a building's 3D blueprint, aiding in structural analysis.

Distance Based on Latitude and Longitude (Haversine): Formula, Derivation, and Detailed Examples

Haversine computes spherical distances. Formula: d = 2R arcsin(√[sin²(Δφ/2) + cosφ₁ cosφ₂ sin²(Δλ/2)]) (R=6371 km). Derivation: From spherical law of cosines; haversin(θ)=sin²(θ/2) avoids singularities at poles, refined by R. W. Sinnott in 1984 for GPS.

Drives navigation apps. In 2025, enhances autonomous shipping routes amid climate shifts.

Step-by-Step Example: NYC (40.7128°N, -74.0060°W) to London (51.5074°N, -0.1278°E):
1. To rad: φ₁=0.699, λ₁=-1.292; φ₂=0.898, λ₂=-0.002.
2. Δφ=0.199, Δλ=1.290; a≈0.547.
3. c=2 arcsin(√0.547)≈1.745; d=6371×1.745≈5570 km.

Point 1 (Lat,Lon)	Point 2 (Lat,Lon)	Distance (km)
40.7128, -74.0060	34.0522, -118.2437	3936
51.5074, -0.1278	40.7128, -74.0060	5570
35.6762, 139.6503	51.5074, -0.1278	9557

Real-World Applications: Uber routing (3936 km NYC-LA). Wildlife tracking migrations.

Tip: Degrees to rad (×π/180); use for <90° separations to minimize errors.
Common Error: Forgetting rad conversion—outputs garbage.
Advanced: Vincenty for ellipsoidal Earth (more precise than sphere).

Example: London (51.5074°N, 0.1278°W) to New York (40.7128°N, 74.0060°W)

The distance is approximately 5,570 km (3,461 miles). This informs flight planning, where the path curves over the Atlantic, differing from flat-map straight lines.

Great Circle Distance (Interactive Map): Formula, Derivation, and Detailed Examples

Visualizes Haversine via markers on OpenStreetMap tiles. Formula: Same as above. Derivation: Great circles are sphere geodesics (shortest paths); map projects via Mercator, but computes true spherical distance.

Empowers travel apps. In 2025, integrates with AR glasses for live geo-overlays.

Step-by-Step Example: NYC to London on map:
1. Click NYC marker, then London.
2. Drag to refine; dashed line appears.
3. Auto-updates: ≈5570 km (3461 miles).

Point 1	Point 2	Distance (km)
NYC	London	5570
Tokyo	NYC	10840
Sydney	London	17000

Real-World Applications: Hiking apps (marker drops=trail dist). Disaster response coord sharing.

Tip: Zoom for precision; drag snaps to streets.
Common Error: Antipodal points (exact opposite)—use Vincenty alt.
Advanced: Orthodromic vs loxodromic for constant heading.

Example: New York to London on Map

Place and drag markers to see the distance update to about 5,570 km (3,461 miles). This simulates real-time route exploration, useful for travel planning or educational demos of Earth's curvature.

These distance methods span flat to curved spaces—from Euclid to Einstein's geodesics. Experiment here to grasp their roles in navigation, simulation, and beyond. For manifolds or relativity, consult advanced texts.