Solid angle surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap. In the diagram this radius is given as: Hence for a unit sphere the solid angle of the spherical cap is given as: When = , the spherical cap becomes a hemisphere having a solid angle 2. The solid angle of the complement of the cone (picture a melon with the cone cut out) is clearly: This is also the solid
Surface-area-to-volume ratio requires changing to a less compact shape. Materials with high surface area to volume ratio (e.g. very small diameter, very porous, or otherwise not compact) react at much faster rates than monolithic materials, because more surface is available to react. Examples include grain dust; while grain isn't typically flammable, grain dust is explosive. Finely ground salt dissolves much more quickly than coarse salt. High surface area to volume ratio provides a strong "driving force" to speed up thermodynamic processes that minimize free energy. The ratio between the surface area and volume of cells and organisms has an enormous impact on