Solid angle surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap. In the diagram this radius is given as: Hence for a unit sphere the solid angle of the spherical cap is given as: When = , the spherical cap becomes a hemisphere having a solid angle 2. The solid angle of the complement of the cone (picture a melon with the cone cut out) is clearly: This is also the solid
Filling area conjecture onto , without increasing its volume. This implies that the replacement has not less volume than the original manifold . Filling area conjecture In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the surfaces that fill a closed curve of given length without introducing shortcuts between its points. Every smooth surface or curve in Euclidean space is a metric space, in which the (intrinsic) distance between two points of is defined as the infimum of the lengths of the curves that go from to "along" . For example, on a closed curve formula_1