Spherical triangle

A spherical triangle is a region on a sphere bounded by three arcs of great circles. Its interior angles sum to more than 180 degrees, with the excess determined by its area.

Great circle: A circle on a sphere whose plane passes through the sphere’s center. Each side of a spherical triangle lies along a great circle.
Spherical law of cosines: An identity relating the sides and angles of a triangle on a sphere. It calculates an unknown side or angle from the other measurements.
Celestial navigation: Navigation that uses observed celestial bodies to determine a position on Earth. Its position calculations use spherical triangles linking observer, pole, and celestial body.
Euclidean triangle: A three-sided polygon in a flat plane with interior angles summing to 180 degrees. Unlike a spherical triangle, its angle sum has no positive curvature excess.
Spherical angle: The angle between two great-circle arcs meeting at a point on a sphere. The triangle’s vertices are measured by angles between its sides.
Spherical law of sines: A relation stating that each side’s sine is proportional to the sine of its opposite angle. It provides a direct side-angle relation for spherical triangles.
Great-circle navigation: Route planning that follows great-circle arcs to obtain shortest paths on a sphere. A route and its endpoints can form a spherical triangle with the poles.
Hyperbolic triangle: A triangle in hyperbolic geometry whose interior angles sum to less than 180 degrees. Its angle deficit contrasts with the spherical triangle’s angle excess.
Spherical distance: The length of the shorter great-circle arc between two points on a sphere. Its side lengths are arc distances, not straight-line chords.
Girard's theorem: A theorem stating that a spherical triangle’s area on a unit sphere equals its angle sum minus π. It links the triangle’s angular excess directly to its area.

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