Spherical geometry is a branch of mathematics that deals with geometric shapes and figures on the surface of a sphere, as opposed to the flat surfaces typically studied in Euclidean geometry. It is a non-Euclidean geometry, meaning that it does not abide by some of the postulates of Euclidean geometry, particularly the parallel postulate.
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Spherical geometry studies lengths and angles measured along the surface of a sphere, rather than through its interior. It is a useful model whenever travel is constrained to a round surface. Its straightest paths are great circle arcs: they are geodesics, even though they appear curved in the surrounding Euclidean space. Two distinct great circles meet at an antipodal pair, so the parallel-line picture of Euclidean geometry does not apply.
For a sphere of radius , write two positions as , with unit vectors. Their shorter surface route has great-circle distancewhere the dot product gives the central angle in radians. The straight chord through the surrounding Euclidean space instead has length . For a quarter-circle separation these are and : specifying which distance we mean has a measurable effect. On the unit sphere, and great-circle distance equals the central angle.
Consider the spherical triangle with vertices , and , joined by the shorter great circle arcs. Each side has length . At each vertex the two outgoing tangent vectors point along the other coordinate axes and are orthogonal, so all three interior angles are . This is the positive coordinate octant of the sphere, not a flat triangle drawn inside it.
For a convex minor spherical triangle with interior angles in radians, the spherical excess formula statesOur octant therefore has area . Independently, eight congruent octants partition the sphere, giving again. Thus angles alone determine area when is fixed, unlike Euclidean geometry, where similar triangles can have arbitrarily different areas. The Gauss-Bonnet theorem explains the connection: the sphere's constant Gaussian curvature times the area equals this excess. For spherical triangles whose area is small compared with , the excess is small, recovering the familiar flat picture.
