Spherical Trigonometry Calculator
Calculate great circle distances, spherical angles, and solve spherical triangles with this comprehensive spherical trigonometry calculator. Essential tool for astronomy, navigation, and geosciences applications.
Calculation Results: Great Circle Distance
| Great Circle Distance | 0.00 km |
|---|---|
| Central Angle | 0.00° |
| Initial Bearing | 0.00° |
| Final Bearing | 0.00° |
About Spherical Trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere.
Great Circle Distance
The great-circle distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles.
The formula for great circle distance using the haversine formula is:
a = sin²(Δφ/2) + cos φ₁ ⋅ cos φ₂ ⋅ sin²(Δλ/2)
c = 2 ⋅ atan2(√a, √(1−a))
d = R ⋅ c
Where φ is latitude, λ is longitude, R is earth's radius (mean radius = 6,371km), and angles are in radians.
Spherical Triangles
A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar triangle, and is sometimes called an Euler triangle.
Key formulas in spherical trigonometry include:
- Law of Cosines: cos a = cos b cos c + sin b sin c cos A
- Law of Sines: sin a / sin A = sin b / sin B = sin c / sin C
- Spherical Excess: E = A + B + C - π (in radians)
- Area: A = R² × E (where R is the sphere's radius)
Applications
Spherical trigonometry has many practical applications:
- Navigation: Calculating shortest routes between points on Earth
- Astronomy: Determining positions of celestial objects
- Geodesy: Measuring and representing the Earth's surface
- Cartography: Creating accurate maps and projections
- Satellite Technology: Calculating orbits and coverage areas
Spherical vs. Plane Trigonometry
Spherical trigonometry differs from plane trigonometry in several important ways:
- The sum of angles in a spherical triangle is always greater than 180°
- Similar spherical triangles are congruent (unlike in plane geometry)
- There are no similar but non-congruent triangles on a sphere
- The sides of a spherical triangle are measured in angular units rather than linear units