Spherical sector
In geometry, a spherical sector, also known as a spherical cone, is a portion of a ball that is bounded by a spherical cap and the cone that connects the centre of the sphere to the boundary of the cap. It is the three-dimensional analogue of the sector of a circle.
Volume
If the radius of the sphere is denoted by and the height of the cap by, the volume of the spherical sector isThis may also be written as
where is half the cone aperture angle, i.e., is the angle between the rim of the cap and the axis direction to the middle of the cap as seen from the sphere center. The limiting case is for approaching 180 degrees, which then describes a complete sphere.
The height, is given by
The volume of the sector is related to the area of the cap by:
Area
The curved surface area of the spherical cap isIt is also
where is the solid angle of the spherical sector in steradians, the SI unit of solid angle. One steradian is defined as the solid angle subtended by a cap area of.
Derivation
The volume can be calculated by integrating the differential volume elementover the volume of the spherical sector,
where the integrals have been separated, because the integrand can be separated into a product of functions each with one dummy variable.
The area can be similarly calculated by integrating the differential spherical area element
over the spherical sector, giving
where is inclination and is azimuth. Notice is a constant. Again, the integrals can be separated.