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The minor sector is shaded in green while the major sector is shaded white. A circular sector, also known as circle sector or disk sector or simply a sector (symbol: β), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, with the smaller area being known as the minor sector and the larger being the major sector. [1]
The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of ): a = R 2 2 ( θ − sin β‘ θ ) {\displaystyle a={\tfrac {R^{2}}{2}}\left(\theta -\sin \theta \right)}
Using radians, the formula for the arc length s of a circular arc of radius r and subtending a central angle of measure π is =, and the formula for the area A of a circular sector of radius r and with central angle of measure π is A = 1 2 θ r 2 . {\displaystyle A={\frac {1}{2}}\theta r^{2}.}
A similar calculation using the area of a circular sector θ = 2A/r 2 gives 1 radian as 1 m 2 /m 2 = 1. [10] The key fact is that the radian is a dimensionless unit equal to 1. In SI 2019, the SI radian is defined accordingly as 1 rad = 1. [11] It is a long-established practice in mathematics and across all areas of science to make use of rad ...
[2] The purpose of the proof is not primarily to convince its readers that β 22 / 7 β (or β 3 + 1 / 7 β ) is indeed bigger than π. Systematic methods of computing the value of π exist. If one knows that π is approximately 3.14159, then it trivially follows that π < β 22 / 7 β , which is approximately 3.142857.
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...
1.1 Radian; 1.2 Arc Length of a Circle; 1.3 Area of Sector of a Circle; 1.4 Application of Circular Measure; 2) Differentiation 2.1 Limit and its Relation to Differentiation; 2.2 The First Derivative; 2.3 The Second Derivative; 2.4 Application of Differentiation; 3) Integration 3.1 Integration as the Inverse of Differentiation; 3.2 Indefinite ...
Another proof that uses triangles considers the area enclosed by a circle to be made up of an infinite number of triangles (i.e. the triangles each have an angle of dπ at the center of the circle), each with an area of β 1 / 2 β · r 2 · dπ (derived from the expression for the area of a triangle: β 1 / 2 β · a · b · sinπ ...