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Spherical trigonometry. The octant of a sphere is a spherical triangle with three right angles. Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles.
The intuitive argument is based upon summing the total sector volume from that of infinitesimal triangular pyramids. Utilizing the pyramid (or cone) volume formula of = ′, where is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and ′ is the height of each pyramid from its base to its apex (at the center ...
In spherical trigonometry, the law of cosines (also called the cosine rule for sides[ 1]) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry . Spherical triangle solved by the law of cosines. Given a unit sphere, a "spherical triangle" on the surface of the sphere ...
Terminology for spherical segments. In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes . It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The surface of the spherical segment (excluding the bases) is called spherical zone .
In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it. Spherical geometry or spherics (from Ancient Greek σφαιρικά) is the geometry of the two- dimensional surface of a sphere [ a] or the n -dimensional surface of higher dimensional spheres .
To find the surface area of the sphere, Archimedes argued that just as the area of the circle could be thought of as infinitely many infinitesimal right triangles going around the circumference (see Measurement of the Circle), the volume of the sphere could be thought of as divided into many cones with height equal to the radius and base on the ...
Frustum. In geometry, a frustum ( Latin for 'morsel'); [ a] ( pl.: frusta or frustums) is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting the solid. In the case of a pyramid, the base faces are polygonal and the side faces are trapezoidal. A right frustum is a right pyramid or a right cone ...
Bonaventura Cavalieri, the mathematician the principle is named after. Cavalieri's principle was originally called the method of indivisibles, the name it was known by in Renaissance Europe. [ 2] Cavalieri developed a complete theory of indivisibles, elaborated in his Geometria indivisibilibus continuorum nova quadam ratione promota ( Geometry ...