Solid angle orientation formula_9, and distance from the viewer as: where the surface area of a sphere is . The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2, is the area of a spherical cap on a unit sphere For small such that , this reduces to the area of a circle . The above is found by computing the following double integral using the unit surface element in spherical coordinates: This formula can also be derived without the use of calculus. Over 2200 years ago Archimedes proved that the
Surface-area-to-volume ratio enough to be thoroughly differentiated although after a billion years or so they became too cool to show anything more than very localized and infrequent volcanic activity, of which none is evident at present. Venus and Earth (r>6,000 km) have sufficiently low surface area-to-volume ratios (roughly half that of Mars and much lower than all other known rocky bodies) so that their heat loss is minimal. Surface-area-to-volume ratio The surface-area-to-volume ratio, also called the surface-to-volume ratio and variously denoted sa/vol or SA:V, is the amount of surface area per unit volume of an object or collection of objects. In chemical