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The ratio of the volume of a sphere to the volume of its circumscribed cylinder is 2:3, as was determined by Archimedes. The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder.
Sphere packing in a cylinder is a three-dimensional packing problem with the objective of packing a given number of identical spheres inside a cylinder of specified diameter and length. For cylinders with diameters on the same order of magnitude as the spheres, such packings result in what are called columnar structures .
The disk-shaped cross-sectional area of the sphere is equal to the ring-shaped cross-sectional area of the cylinder part that lies outside the cone. If one knows that the volume of a cone is (), then one can use Cavalieri's principle to derive the fact that the volume of a sphere is , where is the radius.
Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or interstitial packing. When many sizes of spheres (or a distribution) are available, the problem quickly becomes intractable, but some studies of binary hard spheres (two sizes) are ...
The empty space between spheres varies depending on the type of packing. The amount of empty space is measured in the packing density, which is defined as the ratio of the volume of the spheres to the volume of the total convex hull. The higher the packing density, the less empty space there is in the packing and thus the smaller the volume of ...
An example of a spherical cap in blue (and another in red) In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane.It is also a spherical segment of one base, i.e., bounded by a single plane.
Viviani's curve as intersection of a sphere and a cylinder. In the case = +, the cylinder and sphere are tangential to each other at point (,,). The intersection resembles a figure eight: it is a closed curve which intersects itself. The above parametrization becomes
The surface-area-to-volume ratio has physical dimension inverse length (L −1) and is therefore expressed in units of inverse metre (m −1) or its prefixed unit multiples and submultiples. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm 2 and a volume of 1 cm 3. The surface to volume ratio for this cube is thus
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