Surface of revolution never negative, the area is given by If the curve is described by the function , , then the integral becomes for revolution around the -axis, and for revolution around the "y"-axis (using ). These come from the above formula. For example, the spherical surface with unit radius is generated by the curve , , when ranges over . Its area is therefore For the case of the spherical curve with radius , rotated about the -axis A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area. A basic
Solid angle of a planar triangle is equal to "", for the sum of the four internal solid angles of a tetrahedron as follows: where ranges over all six of the dihedral angles between any two planes that contain the tetrahedral faces OAB, OAC, OBC and ABC. A useful formula for calculating the solid angle subtended by the triangular surface ABC where formula_17 are the vector positions of the vertices A, B and C has been given by Oosterom and Strackee (although the result was known earlier by Euler and Lagrange): where denotes the scalar triple product of the three vectors; When