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 area but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces called "piecewise smooth". Such surfaces consist of finitely many pieces that can be represented in the parametric form with a continuously differentiable function formula_4 The area of an individual piece is defined by the formula Thus the area of "S" is obtained by integrating the length of the normal vector formula_6 to the surface over the appropriate region "D" in the parametric "uv" plane. The