Spherical triangle calculator
A spherical triangle is a triangle drawn on the surface of a sphere, with its three sides being arcs of great circles rather than straight lines. Its sides are measured as angles, in degrees, subtended at the center of the sphere, and unlike a flat triangle its three angles always add up to more than 180 degrees. Spherical triangles are the workhorse of navigation, astronomy and geodesy: they describe the shortest path between two cities, the position of a star, and great-circle distances on the globe. This calculator solves the classic side-side-side problem. You give the three sides a, b and c as angular measures, and it returns the three interior angles A, B and C using the spherical law of cosines, then reports the spherical excess, the amount by which the angle sum exceeds 180 degrees. The excess is directly proportional to the area of the triangle on a unit sphere. Enter your own three sides to solve a navigation or astronomy problem or to check a textbook exercise. The results update as you type. Every figure here is computed deterministically from the spherical law of cosines, shown in full below with a worked example that reconciles exactly to the calculator.
The spherical law of cosines gives each angle: cos A = (cos a minus cos b cos c) / (sin b sin c). For sides a = 60, b = 70, c = 80 degrees the angles are A = 61.57, B = 72.59, C = 89.81 degrees, and the spherical excess is 43.97 degrees.
Spherical law of cosines
cos A = ( cos a - cos b cos c ) / ( sin b sin c )
cos B = ( cos b - cos a cos c ) / ( sin a sin c )
cos C = ( cos c - cos a cos b ) / ( sin a sin b )
spherical excess E = A + B + C - 180 degrees
a, b, c are the angular sides; A, B, C the opposite angles
Each angle is recovered from the side opposite it and the other two sides through the spherical law of cosines. The spherical excess E is the surplus over the flat-triangle angle sum of 180 degrees, and on a unit sphere it equals the area of the triangle in steradians.
Worked example
Solve the spherical triangle with sides a = 60, b = 70 and c = 80 degrees.
- cos A = (cos 60 - cos 70 cos 80) / (sin 70 sin 80) = (0.5 - 0.05939) / 0.92541 = 0.47616, so A = 61.57
- cos B = (cos 70 - cos 60 cos 80) / (sin 60 sin 80) = (0.34202 - 0.08682) / 0.85287 = 0.29922, so B = 72.59
- cos C = (cos 80 - cos 60 cos 70) / (sin 60 sin 70) = (0.17365 - 0.17101) / 0.81380 = 0.00324, so C = 89.81
- Angle sum = 61.57 + 72.59 + 89.81 = 223.97 degrees
- Spherical excess = 223.97 - 180 = 43.97 degrees
The angles are 61.57, 72.59 and 89.81 degrees, with a spherical excess of 43.97 degrees. These are the calculator's default inputs, so the results above match the widget exactly.
Spherical triangle calculator: frequently asked questions
What is a spherical triangle?
A spherical triangle is a region on the surface of a sphere bounded by three arcs of great circles. Its sides are measured as angles at the center of the sphere rather than as lengths, and its three interior angles always sum to more than 180 degrees, the surplus being the spherical excess.
What is the spherical law of cosines?
The spherical law of cosines relates the three sides of a spherical triangle to one of its angles: the cosine of a side equals the product of the cosines of the other two sides plus the product of their sines and the cosine of the included angle. Rearranged, it lets you find each angle from the three sides, which is what this calculator does.
What is the spherical excess?
The spherical excess is the amount by which the sum of the three angles exceeds 180 degrees. It is always positive for a real spherical triangle and, on a unit sphere, it equals the area of the triangle measured in steradians. Girard's theorem makes this relationship exact.
Why do the angles add up to more than 180 degrees?
On a curved surface the rules of flat geometry no longer hold. The positive curvature of a sphere forces the angle sum above 180 degrees, with larger triangles having a larger excess. A triangle covering an entire hemisphere, for example, can have three right angles summing to 270 degrees.
What units does this calculator use?
Both the sides and the angles are in degrees. The sides must each be less than 180 degrees and any two sides must sum to more than the third for a valid triangle. The arithmetic is deterministic and the worked example reconciles exactly to the calculator output.
Official sources
- Mathematical functions and reference definitions: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.