A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus). Properties The sections of the surface of revolution made by planes through the axis are called meridional sections. Any meridional section can be considered to be the generatrix in the plane determined by it and the axis. The sections of the surface of revolution made by planes that are perpendicular to the axis are circles. Some special cases of hyperboloids (of either one or two sheets) and elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular. Area formula If the curve is described by the parametric functions , , with ranging over some interval , and the axis of revolution is the -axis, then the area is given by the integral provided that is never negative between the endpoints and . This formula is the calculus equivalent of Pappus's centroid theorem. The quantity comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity is the path of (the centroid of) this small segment, as required by Pappus' theorem. Likewise, when the axis of rotation is the -axis and provided that is never negative, the area is given by If the continuous curve is described by the function , , then the integral becomes for revolution around the -axis, and for revolution around the y-axis (provided ). 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