Spherical cap
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a great circle), so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere. Volume and surface area The volume of the spherical cap and the area of the curved surface may be calculated using combinations of The radius of the sphere The radius of the base of the cap The height of the cap The polar angle between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap If denotes the latitude in geographic coordinates, then , and . The relationship between and is relevant as long as . For example, the red section of the illustration is also a spherical cap for which . The formulas using and can be rewritten to use the radius of the base of the cap instead of , using the Pythagorean theorem: so that Substituting this into the formulas gives: Deriving the surface area intuitively from the spherical sector volume Note that aside from the calculus based argument below, the area of the spherical cap may be derived from the volume of the spherical sector, by an intuitive argument, as The intuitive argument is based upon summing the total sector volume from that of infinitesimal triangular pyramids. Utilizing the pyramid (or cone) volume formula of , where is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and is the height of each pyramid from its base to its apex (at the center of the sphere). Since each , in the limit, is constant and equivalent to the radius of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and: Deriving the volume and surface area using calculus The volume and area formulas may be derived by examining the rotation of the function for , using the formulas the surface of the rotation for the area and the solid of the revolution for the volume. The area is The derivative of is and hence The formula for the area is therefore The volume is Applications Volumes of union and intersection of two intersecting spheres The volume of the union of two intersecting spheres of radii and is where is the sum of the volumes of the two isolated spheres, and the sum of the volumes of the two spherical caps forming their intersection. If is the distance between the two sphere centers, elimination of the variables and leads to Volume of a spherical cap with a curved base The volume of a spherical cap with a curved base can be calculated