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The base of the cone is a circle of radius 2, with area , while the height is 2, so the area is /. Subtracting the volume of the cone from the volume of the cylinder gives the volume of the sphere: = =. The dependence of the volume of the sphere on the radius is obvious from scaling, although that also was not trivial to make rigorous back then.
On the Sphere and Cylinder (Greek: Περὶ σφαίρας καὶ κυλίνδρου) is a treatise that was published by Archimedes in two volumes c. 225 BCE. [1] It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so. [2]
The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The centroids are at a distance a (in red) from the axis of rotation.. In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of ...
Within the cylinder is the cone whose apex is at the center of one base of the cylinder and whose base is the other base of the cylinder. By the Pythagorean theorem , the plane located y {\displaystyle y} units above the "equator" intersects the sphere in a circle of radius r 2 − y 2 {\textstyle {\sqrt {r^{2}-y^{2}}}} and area π ( r 2 − y ...
An n -ball is a ball in an n -dimensional Euclidean space. The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n -ball of radius R is where is the volume of the unit n -ball, the n -ball of radius 1.
A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request. Archimedes (c. 287 BCE – c. 212 BCE), a colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians.
In all of these arrangements each sphere touches 12 neighboring spheres, [2] and the average density is π 3 2 ≃ 0.74048. {\displaystyle {\frac {\pi }{3{\sqrt {2}}}}\simeq 0.74048.} In 1611, Johannes Kepler conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the Kepler ...
Intersection of a sphere and cone emanating from its center. A spherical sector (blue) A spherical sector. In geometry, a spherical sector, [1] also known as a spherical cone, [2] is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the ...