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
Cylinder . The area of the side is known as the "", . An "open cylinder" does not include either top or bottom elements, and therefore has surface area (lateral area) The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore, where is the diameter of the circular top or bottom. For a given volume, the right circular cylinder with the smallest surface area has . Equivalently, for a given surface area, the right circular cylinder with the largest volume has , that is,