spherical triangle

English

A spherical triangle

Noun

spherical triangle (plural spherical triangles)

1. (geometry, spherical geometry) A triangle, described on the surface of the sphere, whose each side is an arc of some great circle.
   • 1830, Pierce Morton, Geometry, Plane, Solid, and Spherical, in Six Books, Baldwin and Craddock, page 189:

     If the three sides of a spherical triangle be each of them equal to a quadrant, the polar triangle will coincide with it; for each of the angular points will be the pole of the side opposite to it.

   • 1876, Edward Olney, A Treatise on Special Or Elementary Geometry, Sheldon & Company, page 220:

     570. Theorem. The sum of the sides of a spherical triangle may be anything between 0 and a circumference.

   • 1893, Crossley William Crosby Barlow, George Hartley Bryan, Elementary Mathematical Astronomy, W. B. Clive, page v,

     A spherical triangle, like a plane triangle, has six parts, viz., its three sides and its three angles. The sides are generally measured by the angles they subtend, so that the six parts are all expressed as angles.

     Any three parts suffice to determine a spherical triangle, but there are certain "ambiguous cases" when the problem admits of more than one solution. The formulæ required in solving spherical triangles form the subject of Spherical Trigonometry, and are in every case different from the analogous formulæ in Plane Trigonometry. There is this further difference, that a spherical triangle is completely determined if its three angles are given.

   • 2012, Daniel Zwillinger, CRC Standard Mathematical Tables and Formulae, 32nd edition, Taylor & Francis (CRC Press / Chapman & Hall), page 218:

     The angles in a spherical triangle do not have to add up to 180 degrees. It is possible for a spherical triangle to have 3 right angles.

Usage notes

• The length of a side is measured by the angle, in radians, that it subtends. In the case of the unit circle, this measure exactly equals the arc length.
• By convention, each side of a proper spherical triangle is less than $\pi$ $(180^{\circ })$.

Hypernyms

• spherical polygon

Hyponyms

• polar triangle

Translations

Triangle on the surface of a sphere, whose sides are arcs of great circles

• French: please add this translation if you can
• German: Kugeldreieck n
• Italian: please add this translation if you can

See also

• spherical geometry
• spherical trigonometry

Further reading

• Spherical trigonometry on Wikipedia.Wikipedia
• Solution of triangles § Solving spherical triangles on Wikipedia.Wikipedia