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 exchange, bringing oxygen into the blood and releasing carbon dioxide from the blood. Similarly, the small intestine has a finely wrinkled internal surface, allowing the body to absorb nutrients efficiently. Cells can achieve a high surface area to volume ratio with an elaborately convoluted surface, like the microvilli lining the small intestine. Increased surface area can also lead to biological problems. More contact with the environment through the surface of a cell or an organ (relative to its volume) increases loss of water and dissolved substances. High surface area to volume ratios also present problems of temperature control in unfavorable