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  2. On the Sphere and Cylinder - Wikipedia

    en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder

    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]

  3. Spherical cap - Wikipedia

    en.wikipedia.org/wiki/Spherical_cap

    If sphere 2 is very large such that , hence and , which is the case for a spherical cap with a base that has a negligible curvature, the above equation is equal to the volume of a spherical cap with a flat base, as expected.

  4. Sphere - Wikipedia

    en.wikipedia.org/wiki/Sphere

    A sphere (from Greek σφαῖρα, sphaîra) [1] is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. [2]

  5. Cavalieri's principle - Wikipedia

    en.wikipedia.org/wiki/Cavalieri's_principle

    In the 3rd century BC, Archimedes, using a method resembling Cavalieri's principle, [5] was able to find the volume of a sphere given the volumes of a cone and cylinder in his work The Method of Mechanical Theorems. In the 5th century AD, Zu Chongzhi and his son Zu Gengzhi established a similar method to find a sphere's volume. [2]

  6. Volume of an n-ball - Wikipedia

    en.wikipedia.org/wiki/Volume_of_an_n-ball

    The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If A n − 1 (r) is the surface area of an (n − 1)-sphere of radius r, then: = (). Applying this to the above integral gives the expression

  7. n-sphere - Wikipedia

    en.wikipedia.org/wiki/N-sphere

    The formula for the volume of the ⁠ ⁠-ball can be derived from this by integration. Similarly the surface area element of the ⁠ ( n − 1 ) {\displaystyle (n-1)} ⁠ -sphere of radius ⁠ r {\displaystyle r} ⁠ , which generalizes the area element of the ⁠ 2 {\displaystyle 2} ⁠ -sphere, is given by

  8. Spherical shell - Wikipedia

    en.wikipedia.org/wiki/Spherical_shell

    An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness t of the shell: [2] V ≈ 4 π r 2 t , {\displaystyle V\approx 4\pi r^{2}t,}

  9. Spherical sector - Wikipedia

    en.wikipedia.org/wiki/Spherical_sector

    If the radius of the sphere is denoted by r and the height of the cap by h, the volume of the spherical sector is =. This may also be written as V = 2 π r 3 3 ( 1 − cos ⁡ φ ) , {\displaystyle V={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,} where φ is half the cone angle, i.e., φ is the angle between the rim of the cap and the direction ...

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