Sphere

A Sphere

A sphere is a shape in space that is like the surface of a ball. Usually, the words ball and sphere mean the same thing. But in mathematics, a sphere is the surface of a ball, which is given by all the points in three dimensional space that are located at a fixed distance from the center. The distance from the center is called the radius of the sphere. When the sphere is filled in with all the points inside, it is called a ball.

Common things that have the shape of a sphere are basketballs, superballs, and playground balls. The Earth and the Sun are nearly spherical, meaning sphere-shaped.

A sphere is the three-dimensional analog of a circle.

Surface area

Using the circumference: [math]\displaystyle{ A=\frac{c^2}{\pi}=\frac{2c^2}{\tau} }[/math]

Using the diameter: [math]\displaystyle{ A=\pi d^2=\frac{\tau d^2}{2} }[/math]

Using the radius: [math]\displaystyle{ A=2\tau r^2=4\pi r^2 }[/math]

Using the volume: [math]\displaystyle{ A=\sqrt[3]{3\tau V^2}=\sqrt[3]{6\pi V^2} }[/math]

Circumference

Using the surface area: [math]\displaystyle{ c=\sqrt{\pi A}=\sqrt{\frac{\tau A}{2}} }[/math]

Using the diameter: [math]\displaystyle{ c=\pi d=\frac{\tau d}{2} }[/math]

Using the radius: [math]\displaystyle{ c=\tau r=2\pi r }[/math]

Using the volume: [math]\displaystyle{ c=\sqrt[3]{6\pi^2 V}=\sqrt[3]{\frac{3\tau^2 V}{2}} }[/math]

Diameter

Using the surface area: [math]\displaystyle{ d=\sqrt{\frac{A}{\pi}}=\sqrt{\frac{2A}{\tau}} }[/math]

Using the circumference: [math]\displaystyle{ d=\frac{c}{\pi}=\frac{2c}{\tau} }[/math]

Using the radius: [math]\displaystyle{ d=2r }[/math]

Using the volume: [math]\displaystyle{ d=\sqrt[3]{\frac{6V}{\pi}}=\sqrt[3]{\frac{12V}{\tau}} }[/math]

Radius

Using the surface area: [math]\displaystyle{ r=\sqrt{\frac{A}{2\tau}}=\sqrt{\frac{A}{4\pi}} }[/math]

Using the circumference: [math]\displaystyle{ r=\frac{c}{\tau}=\frac{c}{2\pi} }[/math]

Using the diameter: [math]\displaystyle{ r=\frac{d}{2} }[/math]

Using the volume: [math]\displaystyle{ r=\sqrt[3]{\frac{3V}{2\tau}}=\sqrt[3]{\frac{3V}{4\pi}}\approx\frac{\sqrt[3]{15V}}{4} }[/math] (more simple but less precise)

Volume

Using the surface area: [math]\displaystyle{ V=\sqrt{\frac{A^3}{18\tau}}=\sqrt{\frac{A^3}{36\pi}} }[/math]

Using the circumference: [math]\displaystyle{ V=\frac{c^3}{6\pi^2}=\frac{2c^3}{3\tau^2} }[/math]

Using the diameter: [math]\displaystyle{ V=\frac{\pi d^3}{6}=\frac{\tau d^3}{12} }[/math]

Using the radius: [math]\displaystyle{ V=\frac{2\tau r^3}{3}=\frac{4\pi r^3}{3} }[/math]

Equation of a sphere

In Cartesian coordinates, the equation for a sphere with a center at ([math]\displaystyle{ x_0 }[/math], [math]\displaystyle{ y_0 }[/math], [math]\displaystyle{ z_0 }[/math]) is as follows:

[math]\displaystyle{ (x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = r^2 }[/math]

where [math]\displaystyle{ r }[/math] is the radius of the sphere.

Sphere Media

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  Two orthogonal radii of a sphere

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  Sphere and circumscribed cylinder

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  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.

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  Great circle on a sphere

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  Plane section of a sphere: 1 circle

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  Coaxial intersection of a sphere and a cylinder: 2 circles

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  Image of a loxodrome, or rhumb-line, spiraling towards the north pole

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  Kugelspirale

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  General intersection sphere-cylinder

Related pages

• Circle
• Pi
• Tau