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where L and w are, respectively, the perimeter and the width of any curve of constant width. A = π r 2 {\displaystyle A=\pi r^{2}} where A is the area of a circle .
The circumference is 2 π r, and the area of a triangle is half the base times the height, yielding the area π r 2 for the disk. Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , [ 2 ] but did not identify ...
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
By making this assumption, g takes the following form: = (i.e., the direction of g is antiparallel to the direction of r, and the magnitude of g depends only on the magnitude, not direction, of r). Plugging this in, and using the fact that ∂ V is a spherical surface with constant r and area 4 π r 2 {\displaystyle 4\pi r^{2}} ,
The result reported by Charles Hutton (1778) suggested a density of 4.5 g/cm 3 (4 + 1 / 2 times the density of water), about 20% below the modern value. [16] This immediately led to estimates on the densities and masses of the Sun , Moon and planets , sent by Hutton to Jérôme Lalande for inclusion in his planetary tables.
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192 … (sequence A175341 in the OEIS ). Using the same ideas as the usual Gauss circle problem and the fact that the probability that two integers are coprime is 6 / π 2 {\displaystyle 6/\pi ^{2}} , it is relatively straightforward to show that
Specifically, the isoperimetric inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, that 4 π A ≤ L 2 , {\displaystyle 4\pi A\leq L^{2},} and that equality holds if and only if the curve is a circle.