Spheres and theirs geometric figures

The surface of the sphere is the surface generated by a circumference that turns around its diameter. A sphere is the region of the space inside.

The elements of a sphere are:

• The center is the interior point equidistant to any point of the sphere.
• The radius is the distance of the center to a point of the sphere.
• The chord is the segment that joins any two points of the surface.
• The diameter is the chord that passes through the center.
• The poles are the points of the axis that are on the sphere's surface.

In addition to these elements, we can also define:

• The parallels are the circumferences obtained on cutting the sphere's surface with planes perpendicular to the axis of rotation.
• The equator is the circumference obtained on cutting the sphee's surface with the plane perpendicular to the axis of rotation that contains the center of the sphere.
• The meridians are the circumferences obtained on cutting the sphere's surface with planes that contain the rotation axis.

Geometric figures in the sphere

• Hemisphere: It is each of the parts into which the sphere's surface is divided by a plane that passes through the center of the sphere, called diametral plane.

• Semisphere: It is the volume of the hemisphere.

• Spherical crescent: The spherical crescent is the part of the surface of a sphere in between two planes that cut at the diameter of the the sphere.

The area of the spherical crescent is $A=\dfrac{4\cdot \pi \cdot r^2}{360}\cdot n$, where $n$ is the angle between the two planes.

• Spherical wedge: The spherical wedge is the part of a sphere in between two planes that cut though the diameter.

The volume of a spherical wedge is $V=\dfrac{4}{3}\cdot\dfrac{\pi\cdot r^3}{360}\cdot n$, where $n$ is the angle between the two planes.

• Spherical skullcap: A spherical skullcap is each of the parts of the sphere determined by a cutting plane.

The area of the skullcap is $A=2\cdot\pi\cdot R\cdot h$.

The volume of the skullcap is $V=\dfrac{1}{3}\cdot \pi\cdot h^2\cdot (3R-h)$.

• Spherical area: A spherical area is the part of the sphere in between two parallel cutting planes.

The area of the spherical area is $A=2\cdot\pi\cdot R\cdot h$.

The volume of the spherical area is $V=\dfrac{1}{6}\cdot \pi\cdot h\cdot (h^2+3R^2+3 r^2)$.