Spherical Trigonometry Calculator

Calculate spherical triangle sides, angles, great circle distances, and bearings with high precision using multiple algorithms.

Calculation Mode
Great Circle Distance
Spherical Triangle
Initial Bearing
Haversine (Fast)
Vincenty (High Precision)

Haversine Formula: Calculates great-circle distance between two points on a sphere

a = sin²(Δφ/2) + cos φ₁ · cos φ₂ · sin²(Δλ/2)

Preset Examples
NYC → London
LA → Tokyo
Sydney → Santiago
North Pole → Equator
Same Point
Degrees
Radians
Degrees/Minutes/Seconds
Point A Coordinates
Range: -90° to 90° (N positive, S negative)
Range: -180° to 180° (E positive, W negative)
Point B Coordinates
Range: -90° to 90° (N positive, S negative)
Range: -180° to 180° (E positive, W negative)
Calculating...

Understanding Spherical Trigonometry

Spherical trigonometry deals with the relationships between the sides and angles of spherical triangles, which are triangles drawn on the surface of a sphere. Unlike plane triangles, the sides of spherical triangles are arcs of great circles, measured in angular units.

Key Differences from Plane Trigonometry:

  • Sides are measured as angles (degrees or radians), not linear distances
  • The sum of angles in a spherical triangle exceeds 180° (spherical excess)
  • Similarity does not exist: two spherical triangles with equal angles are congruent
  • The area of a spherical triangle is proportional to its spherical excess

Fundamental Formulas

1

Spherical Law of Cosines:

cos a = cos b cos c + sin b sin c cos A

cos A = -cos B cos C + sin B sin C cos a

2

Spherical Law of Sines:

sin a / sin A = sin b / sin B = sin c / sin C

3

Haversine Formula (for distances):

hav(θ) = hav(φ₂ − φ₁) + cos φ₁ cos φ₂ hav(λ₂ − λ₁)

where hav(θ) = sin²(θ/2) = (1 − cos θ)/2

4

Spherical Excess and Area:

E = A + B + C − π (in radians)

Area = R² × E (where R is sphere radius)

Applications

  • Navigation: Calculating great-circle routes for ships and aircraft
  • Astronomy: Determining positions of celestial bodies
  • Geodesy: Measuring distances on Earth's surface
  • Cartography: Creating accurate maps and projections
  • Satellite Communications: Calculating coverage areas and antenna angles

Calculator Features:

  • Calculates great-circle distances using both Haversine and Vincenty formulas
  • Solves spherical triangles using laws of sines and cosines
  • Computes initial and final bearings between two points on a sphere
  • Visualizes results with 2D diagrams and 3D sphere models
  • Supports multiple units (degrees, radians, DMS, km, miles, nautical miles)

Frequently Asked Questions

A great circle is the intersection of a sphere and a plane that passes through the center of the sphere. It represents the shortest path between two points on a sphere's surface. Examples include the equator and lines of longitude (meridians) on Earth.

The haversine formula provides accuracy within about 0.5% for most Earth distances. For higher precision over long distances or near the poles, the Vincenty formula is recommended, which accounts for Earth's ellipsoidal shape and provides accuracy to within 0.5mm.

Spherical excess is the amount by which the sum of angles in a spherical triangle exceeds 180° (or π radians). It's directly proportional to the triangle's area on the sphere: Area = R² × E, where E is the spherical excess in radians and R is the sphere's radius.

Yes, this calculator is suitable for calculating great-circle distances and bearings for navigation. However, for actual navigation, additional factors like magnetic variation, currents, and obstructions must be considered. The calculator assumes a spherical Earth for Haversine and ellipsoidal for Vincenty calculations.

The initial bearing (or forward azimuth) is the direction to follow at the starting point to travel along the great-circle path to the destination. The final bearing is the direction at the destination point back to the starting point. For a great-circle route, the bearing changes continuously along the path (except when following a meridian or the equator).