# Sphere Calculator

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Lets first understand, what is a sphere and how we can find different properties of a sphere?

Sphere

A sphere is a geometrical object in 3-D space that is the surface of a ball.

Like a circle in a 2-D space, a sphere is defined as the set of points that are all at the same distance r from a given point in a 3-D space.

This distance r is the radius of the sphere, which is made up from all points with a distance r from a given point, which is the center of the sphere.

The longest straight line segment connecting two points of the sphere, passes through the center of the sphere and its length is twice the radius of the sphere. It is a diameter of the sphere.

Equation of sphere in 3-D space

A sphere with center \((x_0, y_0, z_0)\) and radius 'r' is the locus of all points \((x, y, z)\) such that

$$(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2$$A sphere of any radius centered at (0,0,0) is an integral surface of the following differential form:

$$x\hspace{0.1cm}dx+y\hspace{0.1cm}dy+z\hspace{0.1cm}dz=0$$This equation reflects that position and velocity vectors of a point (x, y, z) and (dx, dy, dz), traveling on the sphere are always orthogonal to each other.

A sphere can also be constructed as the surface formed by rotating a circle about any of its diameters. Since a circle is a special type of ellipse and a sphere is a special type of ellipsoid of revolution.

Formulas

Let the Radius of the Sphere be \(R\) units.

Enclosed volume of the sphere

Volume of the Sphere will be \(\dfrac{4}{3} \pi R^3 \) cubic units.

The volume inside a sphere is twice the volume between the sphere and the circumscribed cylinder of that sphere, having the height and diameter equal to the diameter of the sphere.

Surface area of the sphere

Surface Area of the Sphere will be \(4\pi R^2\) square units.

Surface area of the sphere is equal to the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius 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 r.

Properties of the sphere

1. The intersection of a sphere and a plane is a circle, a point or empty.

2. The points on the sphere are all the same distance from a fixed point.

3. The contours and plane sections of the sphere are circles.

4. The sphere has constant width and constant girth.

5. All points of a sphere are umbilics.

6. The sphere does not have a surface of centers.

7. The sphere has constant mean curvature.

8. The sphere has constant positive Gaussian curvature.

For calculating different properties of the sphere using above sphere calculator, you have to just write the values in the given input boxes and press the calculate button, you will get the result.