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The van der Waals radius, r w, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. It is named after Johannes Diderik van der Waals, winner of the 1910 Nobel Prize in Physics, as he was the first to recognise that atoms were not simply points and to demonstrate the physical consequences of their size through the van der Waals ...
where A is the area of a squircle with minor radius r, is the gamma function. A = ( k + 1 ) ( k + 2 ) π r 2 {\displaystyle A=(k+1)(k+2)\pi r^{2}} where A is the area of an epicycloid with the smaller circle of radius r and the larger circle of radius kr ( k ∈ N {\displaystyle k\in \mathbb {N} } ), assuming the initial point lies on the ...
In applied sciences, the equivalent radius (or mean radius) is the radius of a circle or sphere with the same perimeter, area, or volume of a non-circular or non-spherical object. The equivalent diameter (or mean diameter ) ( D {\displaystyle D} ) is twice the equivalent radius.
Area#Area formulas – Size of a two-dimensional surface; Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities
Using radians, the formula for the arc length s of a circular arc of radius r and subtending a central angle of measure 𝜃 is =, and the formula for the area A of a circular sector of radius r and with central angle of measure 𝜃 is A = 1 2 θ r 2 . {\displaystyle A={\frac {1}{2}}\theta r^{2}.}
The Wigner–Seitz radius, named after Eugene Wigner and Frederick Seitz, is the radius of a sphere whose volume is equal to the mean volume per atom in a solid (for first group metals). [1] In the more general case of metals having more valence electrons, r s {\\displaystyle r_{\\rm {s}}} is the radius of a sphere whose volume is equal to the ...
The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation:
It is more generally true that the area of the circle of a fixed radius R is a strictly decreasing function of the curvature. In all cases, if k {\displaystyle k} is the curvature (constant, positive or negative), then the isoperimetric inequality for a domain with area A and perimeter L is