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A sphere (from Greek σφαῖρα — ''sphaira'', "globe, ball"(σφαῖρα ), Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus) is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions). Beddoe, Jennifer - (Sphere: Definition & Formulas ) - Study.com. Retrieved 15 July 2015. Like a circle, which geometrically is a two-dimensional object, a sphere is defined mathematically as the set of points that are all at the same distance from a given point, but in three-dimensional space. This distance is the radius of the ball, and the given point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius; it is a diameter of the ball.While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and the ball (a three-dimensional shape that includes the sphere as well as everything inside the sphere). The ball and the sphere share the same radius, diameter, and center. ''==Surface area==Surface area of a sphere is a redirect that points to this section -->The surface area of a sphere is::A = 4\pi r^2.Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder (for example, the Lambert cylindrical equal-area projection) is area-preserving; it equals the derivative of the formula for the volume with respect to because the total volume inside a sphere of radius 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 . At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius is simply the product of the surface area at radius and the infinitesimal thickness.At any given radius , the incremental volume () equals the product of the surface area at radius and the thickness of a shell ()::\delta V \approx A(r) \cdot \delta r. The total volume is the summation of all shell volumes::V \approx \sum A(r) \cdot \delta r.In the limit as approaches zeroPages 141, 149. this equation becomes::V = \int_0^r A(r) \, dr.Substitute ::\frac\pi r^3 = \int_0^r A(r) \, dr.Differentiating both sides of this equation with respect to yields as a function of ::\!4\pi r^2 = A(r).Which is generally abbreviated as::\!A = 4\pi r^2.Alternatively, the area element on the sphere is given in spherical coordinates by . In cartesian coordinates, the area element is: dS=\fracx_^}}\Pi_dx_,\;\forall k. For more generality, see area element.The total area can thus be obtained by integration::A = \int_0^ \int_0^\pi r^2 \sin\theta \, d\theta \, d\varphi = 4\pi r^2.
A sphere (from Greek σφαῖρα — ''sphaira'', "globe, ball"〔(σφαῖρα ), Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus〕) is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions).〔 Beddoe, Jennifer - (Sphere: Definition & Formulas ) - Study.com. Retrieved 15 July 2015.〕 Like a circle, which geometrically is a two-dimensional object, a sphere is defined mathematically as the set of points that are all at the same distance from a given point, but in three-dimensional space. This distance is the radius of the ball, and the given point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius; it is a diameter of the ball.
While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and the ball (a three-dimensional shape that includes the sphere as well as everything inside the sphere). The ball and the sphere share the same radius, diameter, and center. ''
==Surface area==
The surface area of a sphere is:
:
Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder (for example, the Lambert cylindrical equal-area projection) is area-preserving; it equals the derivative of the formula for the volume with respect to because the total volume inside a sphere of radius 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 . At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius is simply the product of the surface area at radius and the infinitesimal thickness.
At any given radius , the incremental volume () equals the product of the surface area at radius and the thickness of a shell ():
:
The total volume is the summation of all shell volumes:
:
In the limit as approaches zero〔Pages 141, 149. 〕 this equation becomes:
:
Substitute :
:
Differentiating both sides of this equation with respect to yields as a function of :
:
Which is generally abbreviated as:
:
Alternatively, the area element on the sphere is given in spherical coordinates by . In cartesian coordinates, the area element is
:
For more generality, see area element.
The total area can thus be obtained by integration:
:
抄文引用元・出典: フリー百科事典『 The surface area of a sphere is::A = 4\pi r^2.Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder (for example, the Lambert cylindrical equal-area projection) is area-preserving; it equals the derivative of the formula for the volume with respect to because the total volume inside a sphere of radius 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 . At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius is simply the product of the surface area at radius and the infinitesimal thickness.At any given radius , the incremental volume () equals the product of the surface area at radius and the thickness of a shell ()::\delta V \approx A(r) \cdot \delta r. The total volume is the summation of all shell volumes::V \approx \sum A(r) \cdot \delta r.In the limit as approaches zeroPages 141, 149. this equation becomes::V = \int_0^r A(r) \, dr.Substitute ::\frac\pi r^3 = \int_0^r A(r) \, dr.Differentiating both sides of this equation with respect to yields as a function of ::\!4\pi r^2 = A(r).Which is generally abbreviated as::\!A = 4\pi r^2.Alternatively, the area element on the sphere is given in spherical coordinates by . In cartesian coordinates, the area element is: dS=\fracx_^}}\Pi_dx_,\;\forall k. For more generality, see area element.The total area can thus be obtained by integration::A = \int_0^ \int_0^\pi r^2 \sin\theta \, d\theta \, d\varphi = 4\pi r^2.">ウィキペディア（Wikipedia）』
■The surface area of a sphere is::A = 4\pi r^2.Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder (for example, the Lambert cylindrical equal-area projection) is area-preserving; it equals the derivative of the formula for the volume with respect to because the total volume inside a sphere of radius 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 . At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius is simply the product of the surface area at radius and the infinitesimal thickness.At any given radius , the incremental volume () equals the product of the surface area at radius and the thickness of a shell ()::\delta V \approx A(r) \cdot \delta r. The total volume is the summation of all shell volumes::V \approx \sum A(r) \cdot \delta r.In the limit as approaches zeroPages 141, 149. this equation becomes::V = \int_0^r A(r) \, dr.Substitute ::\frac\pi r^3 = \int_0^r A(r) \, dr.Differentiating both sides of this equation with respect to yields as a function of ::\!4\pi r^2 = A(r).Which is generally abbreviated as::\!A = 4\pi r^2.Alternatively, the area element on the sphere is given in spherical coordinates by . In cartesian coordinates, the area element is: dS=\fracx_^}}\Pi_dx_,\;\forall k. For more generality, see area element.The total area can thus be obtained by integration::A = \int_0^ \int_0^\pi r^2 \sin\theta \, d\theta \, d\varphi = 4\pi r^2.">ウィキペディアで「A sphere (from Greek σφαῖρα — ''sphaira'', "globe, ball"(σφαῖρα ), Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus) is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions). Beddoe, Jennifer - (Sphere: Definition & Formulas ) - Study.com. Retrieved 15 July 2015. Like a circle, which geometrically is a two-dimensional object, a sphere is defined mathematically as the set of points that are all at the same distance from a given point, but in three-dimensional space. This distance is the radius of the ball, and the given point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius; it is a diameter of the ball.While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and the ball (a three-dimensional shape that includes the sphere as well as everything inside the sphere). The ball and the sphere share the same radius, diameter, and center. ''==Surface area==Surface area of a sphere is a redirect that points to this section -->The surface area of a sphere is::A = 4\pi r^2.Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder (for example, the Lambert cylindrical equal-area projection) is area-preserving; it equals the derivative of the formula for the volume with respect to because the total volume inside a sphere of radius 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 . At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius is simply the product of the surface area at radius and the infinitesimal thickness.At any given radius , the incremental volume () equals the product of the surface area at radius and the thickness of a shell ()::\delta V \approx A(r) \cdot \delta r. The total volume is the summation of all shell volumes::V \approx \sum A(r) \cdot \delta r.In the limit as approaches zeroPages 141, 149. this equation becomes::V = \int_0^r A(r) \, dr.Substitute ::\frac\pi r^3 = \int_0^r A(r) \, dr.Differentiating both sides of this equation with respect to yields as a function of ::\!4\pi r^2 = A(r).Which is generally abbreviated as::\!A = 4\pi r^2.Alternatively, the area element on the sphere is given in spherical coordinates by . In cartesian coordinates, the area element is: dS=\fracx_^}}\Pi_dx_,\;\forall k. For more generality, see area element.The total area can thus be obtained by integration::A = \int_0^ \int_0^\pi r^2 \sin\theta \, d\theta \, d\varphi = 4\pi r^2.」の詳細全文を読む
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