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The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. [11] The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area ...
A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...
The unit sphere is often used as a model for spherical geometry because it has constant sectional curvature of 1, which simplifies calculations. In trigonometry, circular arc length on the unit circle is called radians and used for measuring angular distance; in spherical trigonometry surface area on the unit sphere is called steradians and ...
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.
For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016 [7]) is 2π ⋅ 6371 2 | sin 90° − sin 66.56° | = 21.04 million km 2 (8.12 million sq mi), or 0.5 ⋅ | sin 90° − sin 66.56° | = 4.125% of the total surface area of the Earth.
The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is: [6] A = 4πr 2 (sphere), where r is the radius of the sphere.
A is the surface area of the spherical cap, , r is the radius of the sphere, h is the height of the cap, and; sr is the unit, steradian. Because the surface area A of a sphere is 4πr 2, the definition implies that a sphere subtends 4π steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends 1/4π ≈ 0.07958 of a sphere. By ...
A set of points drawn from a uniform distribution on the surface of a unit 2-sphere, generated using Marsaglia's algorithm. To generate uniformly distributed random points on the unit -sphere (that is, the surface of the unit -ball), Marsaglia (1972) gives the following algorithm.