The sphere, a curved geometric body without edges or vertices, stands out for the uniformity of distances between all its points and the center. According to its definition, this three-dimensional shape is generated by performing a complete rotation of a circle around its diameter, creating a surface of revolution.

A notable property of the sphere is that it has the smallest surface area of all shapes enclosing a specific volume. This characteristic, together with its symmetry and geometric perfection, makes the sphere a fundamental and efficient figure in three-dimensional geometry.

## What Is a Sphere?

A sphere is a perfectly symmetrical and closed three-dimensional geometric figure, whose surface consists of all points equidistant from its center. Characterized by its complete absence of edges and vertices, the sphere exhibits symmetry from any perspective.

Spheres are present in nature and their properties are applied in various scientific fields.

## Features and Properties

Spheres are three-dimensional geometric figures with unique characteristics that distinguish them. Here are some of its most notable properties:

- Symmetry: The spheres exhibit perfect symmetry from any point of view. Any plane passing through its center divides the sphere into two equal halves.
- Curved surface: The surface of a sphere is a continuous curve without edges or vertices. All points on the surface are equidistant from the center.
- Center: Each sphere has a center point from which all distances to the surface are equal.
- Absence of edges and vertices: Unlike polyhedra and other geometric bodies, spheres lack edges and vertices, contributing to their simplicity and uniformity.

## Essential Components of a Sphere

The following elements define the sphere, from its center point to the lines and circumferences that characterize its unique three-dimensional shape:

**Center**: The fixed point on the sphere equidistant from all points on its curved surface. This center is located at the same distance from any surface point.**Axis**: An infinite line that passes through the center of the sphere, providing a directional reference for the geometric body.**Radius**: The distance between the center of the sphere and any point on its surface, defining the radial extent of the three-dimensional solid.**Diameter**: The length of the straight line connecting two points on the surface, passing through the center. Its value is twice the radius, representing the maximum extension of the sphere.**Parallels**: Circles formed by cutting the solid with a plane perpendicular to the axis, creating circular sections.**Meridians**: Circumferences resulting from the section of the sphere by a plane that contains the axis, offering circular sections with specific orientation.**Equator**: The parallel whose center coincides with the center of the sphere, highlighting a special point in its structure.

## Area Calculation

To calculate the surface area of a sphere, the following mathematical formula is used:

A = 4·π·r²

Where

A is the value of the surface area of the sphere. The units of area in the SI measurements are square meters.

r is the radius expressed in meters.

## Volume Formula

To calculate the volume based on the radius of the sphere we can use the following formula:

V = (4·π·r³)/3

Where

V is the volume expressed in cubic meters.

r is the value of the radius expressed in meters.

The volume of the sphere is equal to 2/3 the volume of the circumscribed cylinder in the figure.

## Equation of the Sphere

The general equation of a sphere in a three-dimensional coordinate system is expressed as:

(x−h)²+(y−k)²+(z−l)²=r²

Where:

(h,k,l) are the coordinates of the center of the sphere.

r is the radius of the sphere.

This equation reflects the idea that each point (x,y,z) on the surface of the sphere meets the condition that the sum of the squares of the differences between its coordinates and those of the center is equal to the square of the radius.

When the center of the sphere is at the origin of the coordinate system (0,0,0), the equation simplifies to:

x²+y²+z²=r²

This form of equation defines a sphere centered at the origin with radius r. In both cases, the equation of the sphere is fundamental for the representation and understanding of three-dimensional geometry.

## Spherical Coordinates

Spherical coordinates are a three-dimensional coordinate system used to specify the position of a point in space using two angles and a radial distance from a common origin.

This system is especially useful when working with spherical symmetry problems, such as in physics, astronomy, or engineering.

In spherical coordinates, a point P is defined by three components:

Radius (r): The distance from the origin to point P. It is a non-negative real number.

Colatitude (θ): The angle measured from the positive z axis to the line segment connecting the origin to point P. It varies from 0∘ to 180.

Length (ϕ): The angle measured from the positive x axis in the xy plane to the plane containing point P. It varies from 0∘ to 360∘.

The conversion formulas between Cartesian coordinates (x,y,z) and spherical coordinates (r,θ,ϕ) are:

x = r · sinθ · cosϕ

y = r · sinθ · sinϕ

z = r · cosθ

These coordinates are particularly useful for describing phenomena that exhibit spherical symmetry, such as electromagnetic radiation from an antenna, particle scattering in particle physics, or the position of celestial objects in astronomy.