Great-Circle Distance and Bearing Calculator
Calculate the shortest surface distance and initial bearing between two latitude/longitude points on an entered spherical Earth model.
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Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Form the angle between unit radius vectors using atan2 of their cross-product magnitude and dot product. Multiply the central angle by sphere radius. Resolve the initial tangent direction into local north and east for bearing.
Central angle=atan2(|u×v|,u·v); surface distance=R×angle
Worked example
On a sphere of radius 6371 km, the equatorial route from (0°,0°) to (0°,90°) covers πR/2≈10007.5434 km and starts due east (bearing 90°).
Assumptions and input details
Coordinates use decimal degrees; latitude −90 to 90 and longitude −180 to 180. Radius is explicitly entered, positive 0.001–1e12 m; 6371 km is a convenient spherical approximation. This is a spherical great-circle calculation, not an ellipsoidal WGS84 geodesic, rhumb line, road route or terrain distance. Bearing is clockwise from local true north and generally changes along the path. Initial bearing is undefined at a starting pole, coincident points or antipodes, including within 1e−12 rad of those point-pair singularities. Near antipodes the selected bearing is numerically sensitive. It is not a navigation-grade routing service.
References: Wolfram MathWorld great circles
Unit definitions
Unit selection is independent of language. Each selector contains only units for its physical quantity. Calculations use defined base units internally.
Choose from 23 length definitions, including explicitly labelled regional measures. Area conversions square the length factor; geometric volumes cube it.
Regional measures are not interchangeable merely because their names look similar. This is a documented selection, not a claim to include every historical unit.
NIST measurement tables · Japanese traditional units · Mainland China and Hong Kong unit distinctions
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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