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
Steinmetz solid bicylinder describes a total of 8 pyramids. The volume of the cube (red) minus the volume of the eight pyramids (blue) is the volume of the bicylinder (white). The volume of the 8 pyramids is: formula_12, and then we can calculate that the bicylinder volume is formula_13 The intersection of three cylinders whith perpendicularly intersecting axes generates a surface of a solid with vertices where 3 edges meet and vertices where 4 edges meet. The set of vertices can be considered as the edges of a rhombic dodecahedron. The key for the determination of volume and surface area is the