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
Surface integral of the graph of some scalar function, say formula_3, we have where formula_5. So that formula_6, and formula_7. So, which is the standard formula for the area of a surface described this way. One can recognize the vector in the second line above as the normal vector to the surface. Note that because of the presence of the cross product, the above formulas only work for surfaces embedded in three-dimensional space. This can be seen as integrating a Riemannian volume form on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface. Consider