Spherical Triangle Calculator
Enter three known sides or angles. The Spherical Triangle Calculator uses spherical trigonometry to solve for unknown sides, angles, spherical excess, and area on a sphere.
About the Spherical Triangle Calculator
The Spherical Triangle Calculator is a comprehensive, mathematically precise tool that solves triangles on the surface of a sphere using the principles of spherical trigonometry. This Spherical Triangle Calculator implements the spherical law of cosines and law of sines to accurately determine unknown sides, angles, spherical excess, and surface area. Proudly supported by Agri Care Hub.
Key Formulas in Spherical Trigonometry
Spherical Law of Cosines (for angles): cos C = −cos A cos B + sin A sin B cos c
Spherical Law of Sines: sin a / sin A = sin b / sin B = sin c / sin C
Spherical Excess: E = A + B + C − 180°
Area of spherical triangle: Area = E × π R² / 180° (E in degrees)
Why This Calculator Is Essential
Spherical triangles are fundamental in many fields because the Earth and celestial sphere are spherical. Unlike plane triangles, the sum of angles in a spherical triangle exceeds 180°, and sides are measured as central angles. This calculator is essential for: • Great circle navigation in aviation and maritime routes • Astronomy and celestial coordinate calculations • Geodesy and global positioning systems • Satellite orbit determination and communication • Meteorology and global climate modeling • Surveying large-scale land areas accurately The shortest path on a sphere is a great circle, making spherical trigonometry indispensable for global-scale problems.
How to Use the Spherical Triangle Calculator
- Enter three known values: sides (a, b, c) as central angles in degrees, or angles (A, B, C) in degrees.
- For area calculation, enter the sphere radius R (default is Earth's mean radius 6371 km).
- Leave unknown values blank.
- Click “Solve Spherical Triangle”.
- View all solved sides, angles, spherical excess, and area.
When Should You Use This Tool?
- Calculating great circle distances and bearings between cities
- Aviation flight planning (shortest routes)
- Maritime navigation and sailing
- Astronomy (positions on the celestial sphere)
- GPS and satellite positioning
- Global mapping and GIS applications
- Teaching spherical geometry and trigonometry
Scientific Foundation
Spherical trigonometry dates back to ancient Greek astronomers like Hipparchus and Ptolemy, and was extensively developed during the Islamic Golden Age for astronomical calculations. The formulas are derived from vector dot products on the sphere and are exact for spherical geometry. The spherical excess theorem (Girard’s theorem) relates the area directly to the excess angle sum. Full theory at Spherical Triangle on Wikipedia and classic texts like Todhunter’s *Spherical Trigonometry*.
Conclusion
The Spherical Triangle Calculator brings professional spherical trigonometry to your browser — with perfect mathematical accuracy and beautiful design. Whether you’re a student learning great circle navigation, a pilot planning routes, an astronomer calculating celestial positions, or a geodesist working on global models, this tool delivers precise results every time. For more advanced math and science tools, visit Agri Care Hub.