Spherical Triangle Area Calculator
Calculate spherical excess, area and vertex angles from three minor-arc central side angles.
YOUR CALCULATION
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
Use l’Huilier’s formula for a nondegenerate convex minor-arc spherical triangle.
Excess=4 atan sqrt(tan(s/2)tan((s−a)/2)tan((s−b)/2)tan((s−c)/2)); area=excess R²
Worked example
Three90° sides enclose an octant with excessπ/2 and areaπR²/2.
Assumptions and input details
Triangle inequalities must be strict and perimeter below360°. Ideal sphere, not ellipsoidal geodesy; sides are central angles rather than planar lengths. Degenerate/tiny poorly represented excess is outside numeric scope.
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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