Editing Spherical geometry (section)

{{General geometry}}
{{Short description|Geometry of the surface of a sphere}}
[[Image:Triangles (spherical geometry).jpg|thumb|right|300px|The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees.]]
[[file:Spherical_triangle_3d.png|thumb|right|300px|A sphere with a spherical triangle on it.]]

'''Spherical geometry''' or '''spherics'''  ({{etymology|grc|{{lang|grc|σφαιρικά}}}}) is the [[geometry]] of the two-[[dimension]]al surface of a [[sphere]]{{efn|In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "[[ball (mathematics)|ball]]" (or "solid sphere") are used for the surface together with its 3-dimensional interior.}} or the {{mvar|n}}-dimensional surface of [[n-sphere|higher dimensional spheres]].

Long studied for its practical applications to [[astronomy]], [[navigation]], and [[geodesy]], spherical geometry and the metrical tools of [[spherical trigonometry]] are in many respects analogous to [[Euclidean geometry|Euclidean plane geometry]] and [[trigonometry]], but also have some important differences.

The sphere can be studied either ''extrinsically'' as a surface embedded in 3-dimensional [[Euclidean space]] (part of the study of [[solid geometry]]), or ''intrinsically'' using methods that only involve the surface itself without reference to any surrounding space.