Sphere

A Sphere

A sphere is a round, three-dimensional shape. All points on the edge of the sphere are at the same distance from the center. The distance from the center is called the radius of the sphere. A real-world sphere is called a globe if it is large (such as the Earth), and as a ball if it is small, like an association football.

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

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

Calculating measures of a sphere

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 d2 }[/math]

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

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,y_0,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

Two orthogonal radii of a sphere
Sphere and circumscribed cylinder
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.
Great circle on a sphere
Plane section of a sphere: one circle
Coaxial intersection of a sphere and a cylinder: two circles
Image of a loxodrome, or rhumb-line, spiraling towards the north pole
Clelia spiral with $c = 8$
General intersection sphere-cylinder