Editing Sphere (section)

==Properties==
===Enclosed volume{{anchor|Volume}}===
[[File:Sphere and circumscribed cylinder.svg|thumb|upright=1.1|Sphere and circumscribed cylinder]]

In three dimensions, the [[volume]] inside a sphere (that is, the volume of a [[ball (mathematics)|ball]], but classically referred to as the volume of a sphere) is
:<math>V = \frac{4}{3}\pi r^3 = \frac{\pi}{6}\ d^3 \approx 0.5236 \cdot d^3</math>
where {{mvar|r}} is the radius and {{mvar|d}} is the diameter of the sphere. [[Archimedes]] first derived this formula (''[[On the Sphere and Cylinder]]'' c. 225 BCE) by showing that the volume inside a sphere is twice the volume between the sphere and the [[circumscribe]]d [[cylinder (geometry)|cylinder]] of that sphere (having the height and diameter equal to the diameter of the sphere).<ref>{{harvnb|Steinhaus|1969|loc=p. 223}}.</ref> This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying [[Cavalieri's principle]].<ref>{{cite web|url=http://mathcentral.uregina.ca/QQ/database/QQ.09.01/rahul1.html|title=The volume of a sphere – Math Central|website=mathcentral.uregina.ca|access-date=2019-06-10}}</ref> This formula can also be derived using [[integral calculus]] (i.e., [[disk integration]]) to sum the volumes of an [[infinite number]] of [[Circle#Properties|circular]] disks of infinitesimally small thickness stacked side by side and centered along the {{mvar|x}}-axis from {{math|1=''x'' = −''r''}} to {{math|1=''x'' = ''r''}}, assuming the sphere of radius {{mvar|r}} is centered at the origin.
{{Collapse top|title=Proof of sphere volume, using calculus}}
At any given {{mvar|x}}, the incremental volume ({{mvar|δV}}) equals the product of the cross-sectional [[area of a disc#Onion proof|area of the disk]] at {{mvar|x}} and its thickness ({{mvar|δx}}):
:<math>\delta V \approx \pi y^2 \cdot \delta x.</math>

The total volume is the summation of all incremental volumes:
:<math>V \approx \sum \pi y^2 \cdot \delta x.</math>

In the limit as {{mvar|δx}} approaches zero,<ref name="delta"/> this equation becomes:
:<math>V = \int_{-r}^{r} \pi y^2 dx.</math>

At any given {{mvar|x}}, a right-angled triangle connects {{mvar|x}}, {{mvar|y}} and {{mvar|r}} to the origin; hence, applying the [[Pythagorean theorem]] yields:
:<math>y^2 = r^2 - x^2.</math>

Using this substitution gives
:<math>V = \int_{-r}^{r} \pi \left(r^2 - x^2\right)dx,</math>

which can be evaluated to give the result
:<math>V = \pi \left[r^2x - \frac{x^3}{3} \right]_{-r}^{r} = \pi \left(r^3 - \frac{r^3}{3} \right) - \pi \left(-r^3 + \frac{r^3}{3} \right) = \frac43\pi r^3.</math>

An alternative formula is found using [[spherical coordinates]], with [[volume element]]
:<math> dV=r^2\sin\theta\, dr\, d\theta\, d\varphi</math>
so
:<math>V=\int_0^{2\pi} \int_{0}^{\pi} \int_0^r r'^2\sin\theta\, dr'\, d\theta\, d\varphi
= 2\pi \int_{0}^{\pi} \int_0^r r'^2\sin\theta\, dr'\, d\theta
= 4\pi \int_0^r r'^2\, dr'\ =\frac43\pi r^3.</math>
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For most practical purposes, the volume inside a sphere [[Inscribed figure|inscribed]] in a cube can be approximated as 52.4% of the volume of the cube, since {{math|1=''V'' = {{sfrac|{{pi}}|6}} ''d''<sup>3</sup>}}, where {{mvar|d}} is the diameter of the sphere and also the length of a side of the cube and {{sfrac|{{pi}}|6}}&nbsp;≈&nbsp;0.5236. For example, a sphere with diameter 1&nbsp;m has 52.4% the volume of a cube with edge length 1{{Spaces}}m, or about 0.524&nbsp;m<sup>3</sup>.

===Surface area{{anchor|Area}}===
<!--[[Surface area of a sphere]] is a redirect that points to this section.-->
The [[surface area]] of a sphere of radius {{mvar|r}} is:
:<math>A = 4\pi r^2.</math>

[[Archimedes]] first derived this formula<ref name=MathWorld_Sphere>{{MathWorld |title=Sphere |id=Sphere}}</ref> from the fact that the projection to the lateral surface of a [[circumscribe]]d cylinder is area-preserving.<ref>{{harvnb|Steinhaus|1969|loc=p. 221}}.</ref> Another approach to obtaining the formula comes from the fact that it equals the [[derivative]] of the formula for the volume with respect to {{mvar|r}} because the total volume inside a sphere of radius {{mvar|r}} can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius {{mvar|r}}. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius {{mvar|r}} is simply the product of the surface area at radius {{mvar|r}} and the infinitesimal thickness.

{{Collapse top|title=Proof of surface area, using calculus}}
At any given radius {{mvar|r}},{{NoteTag |{{mvar|r}} is being considered as a variable in this computation.}} the incremental volume ({{mvar|δV}}) equals the product of the surface area at radius {{mvar|r}} ({{math|''A''(''r'')}}) and the thickness of a shell ({{mvar|δr}}):
:<math>\delta V \approx A(r) \cdot \delta r. </math>

The total volume is the summation of all shell volumes:
:<math>V \approx \sum A(r) \cdot \delta r.</math>

In the limit as {{mvar|δr}} approaches zero<ref name="delta">{{cite book
|author1=E.J. Borowski |author2=J.M. Borwein |title=Collins Dictionary of Mathematics
|year=1989 |isbn=978-0-00-434347-1|pages=141, 149|publisher=Collins }}</ref> this equation becomes:
:<math>V = \int_0^r A(r) \, dr.</math>

Substitute {{mvar|V}}:
:<math>\frac43\pi r^3 = \int_0^r A(r) \, dr.</math>

Differentiating both sides of this equation with respect to {{mvar|r}} yields {{mvar|A}} as a function of {{mvar|r}}:
:<math>4\pi r^2 = A(r).</math>

This is generally abbreviated as:
:<math>A = 4\pi r^2,</math>
where {{mvar|r}} is now considered to be the fixed radius of the sphere.

Alternatively, the [[area element]] on the sphere is given in [[spherical coordinates]] by {{math|1=''dA'' = ''r''<sup>2</sup> sin ''θ dθ dφ''}}. The total area can thus be obtained by [[Integral|integration]]:
:<math>A = \int_0^{2\pi} \int_0^\pi r^2 \sin\theta \, d\theta \, d\varphi = 4\pi r^2.</math>
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The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area.<ref>{{cite journal |last1=Osserman |first1=Robert |journal=Bulletin of the American Mathematical Society |title=The isoperimetric inequality |date=1978 |volume=84 |issue=6 |page=1187 |doi=10.1090/S0002-9904-1978-14553-4 |url=https://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/ |access-date=14 December 2019 |ref=Osserman|doi-access=free }}</ref> The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the [[surface tension]] locally minimizes surface area.

The surface area relative to the mass of a ball is called the [[specific surface area]] and can be expressed from the above stated equations as
:<math>\mathrm{SSA} = \frac{A}{V\rho} = \frac{3}{r\rho}</math>
where {{mvar|ρ}} is the [[density]] (the ratio of mass to volume).

===Other geometric properties===
A sphere can be constructed as the surface formed by rotating a [[circle]] one half revolution about any of its [[diameter]]s; this is very similar to the traditional definition of a sphere as given in [[Euclid's Elements]]. Since a circle is a special type of [[ellipse]], a sphere is a special type of [[ellipsoid of revolution]]. Replacing the circle with an ellipse rotated about its [[major axis]], the shape becomes a prolate [[spheroid]]; rotated about the minor axis, an oblate spheroid.<ref>{{harvnb|Albert|2016|loc=p. 60}}.</ref>

A sphere is uniquely determined by four points that are not [[coplanar]]. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc.<ref>{{harvnb|Albert|2016|loc=p. 55}}.</ref> This property is analogous to the property that three [[collinear|non-collinear]] points determine a unique circle in a plane.

Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.

By examining the [[Circle of a sphere#Sphere-sphere intersection|common solutions of the equations of two spheres]], it can be seen that two spheres intersect in a circle and the plane containing that circle is called the '''radical plane''' of the intersecting spheres.<ref>{{harvnb|Albert|2016|loc=p. 57}}.</ref> Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).<ref name=Woods267>{{harvnb|Woods|1961|loc=p. 267}}.</ref>

The angle between two spheres at a real point of intersection is the [[dihedral angle]] determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection.<ref>{{harvnb|Albert|2016|loc=p. 58}}.</ref> They intersect at right angles (are [[Orthogonality|orthogonal]]) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.<ref name=Woods266 />

====Pencil of spheres====
{{Main|Pencil (mathematics)#Pencil of spheres}}
If {{math|1=''f''(''x'', ''y'', ''z'') = 0}} and {{math|1=''g''(''x'', ''y'', ''z'') = 0}} are the equations of two distinct spheres then
:<math>s f(x,y,z) + t g(x,y,z) = 0</math>
is also the equation of a sphere for arbitrary values of the parameters {{mvar|s}} and {{mvar|t}}. The set of all spheres satisfying this equation is called a '''pencil of spheres''' determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.<ref name=Woods266 />

===Properties of the sphere===
[[File:Sphere section.png|thumb|A normal vector to a sphere, a normal plane and its normal section. The curvature of the curve of intersection is the sectional curvature. For the sphere each normal section through a given point will be a circle of the same radius: the radius of the sphere. This means that every point on the sphere will be an umbilical point.]]

In their book ''Geometry and the Imagination'', [[David Hilbert]] and [[Stephan Cohn-Vossen]] describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere.<ref>{{cite book
|author1=Hilbert, David
|author-link=David Hilbert
|author2=Cohn-Vossen, Stephan
|title=Geometry and the Imagination
|url=https://archive.org/details/geometryimaginat00davi_0|url-access=registration|edition=2nd
|year=1952
|publisher=Chelsea
|isbn=978-0-8284-1087-8|chapter=Eleven properties of the sphere|pages=215–231}}</ref> Several properties hold for the [[plane (mathematics)|plane]], which can be thought of as a sphere with infinite radius. These properties are:
#''The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.''
#: The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar [[Circle#Circle of Apollonius|result]] of [[Apollonius of Perga]] for the [[circle]]. This second part also holds for the [[plane (mathematics)|plane]].
#''The contours and plane sections of the sphere are circles.''
#: This property defines the sphere uniquely.
#''The sphere has constant width and constant girth.''
#: The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the [[Meissner body]]. The girth of a surface is the [[circumference]] of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.
#''All points of a sphere are [[umbilic]]s.''
#: At any point on a surface a [[Normal (geometry)|normal direction]] is at right angles to the surface because on the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane that contains the normal with the surface will form a curve that is called a ''normal section,'' and the curvature of this curve is the ''normal curvature''. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the [[principal curvature]]s. Any closed surface will have at least four points called ''[[umbilical point]]s''. At an umbilic all the sectional curvatures are equal; in particular the [[principal curvature]]s are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.
#: For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.
#''The sphere does not have a surface of centers.''
#: For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along on the normal line. For example, the two centers corresponding to the maximum and minimum sectional curvatures are called the ''focal points'', and the set of all such centers forms the [[focal surface]].
#: For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special:
#: * For [[channel surface]]s one sheet forms a curve and the other sheet is a surface
#: * For [[Cone (geometry)|cones]], cylinders, [[torus|tori]] and [[Dupin cyclide|cyclides]] both sheets form curves.
#: * For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere.
#''All geodesics of the sphere are closed curves.''
#: [[Geodesics]] are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. Many other surfaces share this property.
#''Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.''
#: It follows from [[isoperimetric inequality]]. These properties define the sphere uniquely and can be seen in [[soap bubble]]s: a soap bubble will enclose a fixed volume, and [[surface tension]] minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape). It can also be seen in planets and stars where gravity minimizes surface area for large celestial bodies.
#''The sphere has the smallest total mean curvature among all convex solids with a given surface area.''
#: The [[mean curvature]] is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere.
#''The sphere has constant mean curvature.''
#: The sphere is the only [[Embedding|embedded]] surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as [[minimal surface]]s have constant mean curvature.
#''The sphere has constant positive Gaussian curvature.''
#: [[Gaussian curvature]] is the product of the two principal curvatures. It is an intrinsic property that can be determined by measuring length and angles and is independent of how the surface is [[embedding|embedded]] in space. Hence, bending a surface will not alter the Gaussian curvature, and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries, and the sphere is the only surface that lacks a boundary with constant, positive Gaussian curvature. The [[pseudosphere]] is an example of a surface with constant negative Gaussian curvature.
#''The sphere is transformed into itself by a three-parameter family of rigid motions.''
#: Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see [[Euler angles]]). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the [[rotation group SO(3)]]. The plane is the only other surface with a three-parameter family of transformations (translations along the {{mvar|x}}- and {{mvar|y}}-axes and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the [[Surface of revolution|surfaces of revolution]] and [[helicoid]]s are the only surfaces with a one-parameter family.