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
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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.
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Semisphere: It is the volume of the hemisphere.
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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.
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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.
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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 $$V=\dfrac{1}{3}\cdot \pi\cdot h^2\cdot (3R-h)$$.
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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 $$V=\dfrac{1}{6}\cdot \pi\cdot h\cdot (h^2+3R^2+3 r^2)$$.