Solid angle of its faces is one-sixth of that, or sr. Solid angles can also be measured in square degrees (1 sr = () square degrees), in square minutes and square seconds, or in fractions of the sphere (1 sr = fractional area), also known as spat (1 sp = 4 sr). In spherical coordinates there is a formula for the differential, where is the colatitude (angle from the North pole) and is the longitude. The solid angle for an arbitrary oriented surface S subtended at a point P is equal to the solid angle of the projection of the surface S
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