Spherical Triangle Calculator
Solve spherical triangles on the surface of a sphere given two arc sides and the included angle. Used in celestial navigation, geodesy, and great-circle route planning.
About this calculator
A spherical triangle is formed by three great-circle arcs on a sphere. Its sides (a, b, c) are measured as central angles in degrees or radians, not linear distances. The spherical law of cosines gives the third side: cos(c) = cos(a)·cos(b) + sin(a)·sin(b)·cos(C), where C is the included angle. Once c is known, the remaining angles can be found using: cos(A) = (cos(a) − cos(b)·cos(c)) / (sin(b)·sin(c)). The spherical excess E = A + B + C − π gives the area of the triangle on a unit sphere (area = E·R², where R is the sphere's radius). Unlike plane triangles, the angle sum in a spherical triangle always exceeds 180°.
How to use
Given two arc sides a = 60° and b = 45°, with included angle C = 80°, find side c. Convert to radians: a ≈ 1.0472, b ≈ 0.7854, C ≈ 1.3963. Apply the spherical law of cosines: cos(c) = cos(60°)·cos(45°) + sin(60°)·sin(45°)·cos(80°) = 0.5 × 0.7071 + 0.8660 × 0.7071 × 0.1736 ≈ 0.3536 + 0.1063 = 0.4599. Therefore c = arccos(0.4599) ≈ 62.6°. Enter those values, choose degrees, and select 'Find Side C'.
Frequently asked questions
How is a spherical triangle different from a flat plane triangle?
In a plane triangle, the three angles always sum to exactly 180°. In a spherical triangle, the angles sum to more than 180°, and the excess above 180° is called the spherical excess, which is proportional to the triangle's area on the sphere. The sides of a spherical triangle are arcs of great circles and are measured in angular units, not meters. The larger the triangle relative to the sphere, the more pronounced these differences become, which is why flat-Earth geometry fails over long navigation distances.
When is the spherical triangle calculator used in real navigation?
Spherical triangle calculations underpin great-circle navigation, which gives the shortest path between two points on Earth's surface. Pilots and ship navigators use spherical trigonometry to compute initial headings and total distances for intercontinental routes. The calculator is also essential in celestial navigation, where the astronomical triangle — formed by the zenith, the celestial pole, and a star — is solved to determine a vessel's position. GPS and modern avionics still rely on these spherical geometry principles internally.
What is the spherical excess and how does it relate to the area of a spherical triangle?
The spherical excess E is defined as E = A + B + C − π (in radians), where A, B, and C are the interior angles of the spherical triangle. For a unit sphere (radius = 1), the area of the triangle equals E in steradians. For a sphere of radius R, the area is E × R². This elegant result, known as Girard's theorem, means that simply measuring the three angles of a spherical triangle is enough to compute its surface area, without needing any side lengths.