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This yields as a special case the well-known formula for the area of a triangle, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point. The 7th-century Indian mathematician Bhāskara I derived the following formula for the area of a trapezoid with consecutive sides a, c, b, d:
The basic quantities describing a sphere (meaning a 2-sphere, a 2-dimensional surface inside 3-dimensional space) will be denoted by the following variables r {\displaystyle r} is the radius, C = 2 π r {\displaystyle C=2\pi r} is the circumference (the length of any one of its great circles ),
The formula was described by Albrecht Ludwig Friedrich Meister (1724–1788) in 1769 [4] and is based on the trapezoid formula which was described by Carl Friedrich Gauss and C.G.J. Jacobi. [5] The triangle form of the area formula can be considered to be a special case of Green's theorem.
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 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.
The area of the trapezoid can be calculated to be half the area of the square, that is 1 2 ( b + a ) 2 . {\displaystyle {\frac {1}{2}}(b+a)^{2}.} The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of 1 2 {\displaystyle {\frac {1}{2}}} , which is removed by multiplying by two ...
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The formula for the area of a trapezoid can be simplified using Pitot's theorem to get a formula for the area of a tangential trapezoid. If the bases have lengths a, b, and any one of the other two sides has length c, then the area K is given by the formula [2] (This formula can be used only in cases where the bases are parallel.)