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On the surface of a sphere, the closest analogue to straight lines are great circles, i.e. circles whose center coincide with the center of the sphere. (For examples, meridians and the equator are great circles on the Earth.) As lines on a plane, great circles on a sphere are the closest connection of two points (if you constrain yourself to lines on the sphere). (cf. geodesic)

An area on the sphere which is bounded by arcThe term arc may refer to: A part of a circle's circumference (also called a circle segment . Minute of arc or second of arc — an angle measure Edge — in graph theory . arc files Arc language Arc lamp Voltaic arc Arc programming language Arch Story arc Ses of great circles is called a spherical polygon. Note that, unlike the case on a plane, spherical "biangles" (two-sided analogs to triangle) are possible (think about peeling an orange).

The sideSide (mod. Eski Adalia , was an ancient city on the Pamphylian coast about 12 miles east of the mouth of the Eurymedon. Possessing a good harbour in the days of small craft, it was the most important place in Pamphylia. Alexander visited and occupied it,s of these polygons is most conveniently specified not by their length, but by the angle, under which its endpoints appear when looked at from the sphere's center. Note that this arc angle, measured in radianIn mathematics and physics, the radian is a unit of angle measure. It is the SI derived unit of angle. It is defined as the angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle. Angle measures in, and multiplied by the sphere's radiusThe word radius ( Latin for "wheel spoke"; plural radii pronounced ray dee-eye has several meanings in English: In classical geometry, a radius of a circle or sphere is any line segment with one endpoint on the circle (i. the circular boundary) and the ot, is the arc length.

Hence, a spherical triangle is specified as usual by its corner angles and its sides, but the sides are not given by their length, but by their arc angle.

Remarkably, the sum of the corner angle is not 180°, as in a planar triangle, but always larger. This surplus is called the spherical excess E: E = α + β + γ − 180°. It allows calculation of the surface area A surrounded by the triangle, which is simply given by A = R² · E (where R is the radius of the sphere). (In other words: E is the solid angleA solid angle is the three dimensional analog of the ordinary angle. Instead of two lines meeting at a vertex, though, one needs a three dimensional figure that meets at a point. Simple examples of objects that do this are a cone or a pyramid. The SI unit (as measured in steradianThe steradian ste from Greek stereos solid) is the SI derived unit of solid angle, and the 3- dimensional equivalent of the radian. The symbol is sr . The steradian is defined as "the solid angle subtended at the center of a sphere of radius r by a portio) spanned up by the triangle.) This formula is an application of the Gauss-Bonnet theoremThe Gauss- Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). Suppose is a compact two-dimensional orient.

To solve a geometric problem on the sphere, one dissects the relevant figure into right spherical triangles (i.e.: one of the triangles corner angles is 90°) because one can then use Neper's pentagon:

Neper's pentagon (also known as Neper's circle) is mnemonic aid to easily find all relations between the angles in a right spherical triangle:

Write the six angles of the triangle in the form of a circle, sticking to the order as they appear in the triangle (i.e.: start with a corner angle, write the arc angle of an attached side next to it, proceed with the next corner angle, etc. and close the circle). Then cross out the 90° corner angle and replace the arc angles adjacend to it by their complement to 90° (i.e. replace, say, a by 90° - a). The five numbers that you now have on your paper form Neper's Pentagon (or Neper's Circle). For them, it holds that the cosine of each angle is equal to

• the product of the cotangents of the angles written next to it, and is also equal to
• the product of the sines of the two angles written opposed to it.

See also the Haversine formula, which relates the lengths of sides and angles in spherical triangles in a numerically stable form for navigation.