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 by considering two spheres with radii and , separated by some distance , and for which their surfaces intersect at . That is, the curvature of the base comes from sphere 2. The volume is thus the difference between sphere 2's cap (with height ) and sphere 1's cap (with height ), This formula is valid only for configurations that satisfy and . If sphere 2 is very large such that , hence and , which is the case for a spherical cap with a base that has a negligible curvature, the above equation is equal to the volume of a spherical cap with a flat base, as expected. Areas of intersecting spheres Consider two intersecting spheres of radii and , with their centers separated by distance . They intersect if From the law of cosines, the polar angle of the spherical cap on the sphere of radius is Using this, the surface area of the spherical cap on the sphere of radius is Surface area bounded by parallel disks The curved surface area of the spherical segment bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius , and caps with heights and , the area is or, using geographic coordinates with latitudes and , For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016) is 2·63712|sin 90° − sin 66.56°| = 21.04 million km2, or 0.5·|sin 90° − sin 66.56°| = 4.125% of the total surface area of the Earth. This formula can also be used to demonstrate that half the surface area of the Earth lies between latitudes 30° South and 30° North in a spherical