Surface of Revolution

A surface of revolution is a surface generated by rotating a two-dimensional curve about an axis. The resulting surface therefore always has azimuthal symmetry. Examples of surfaces of revolution include the apple surface, cone (excluding the base), conical frustum (excluding the ends), cylinder (excluding the ends), Darwin-de Sitter spheroid, Gabriel's horn, hyperboloid, lemon surface, oblate spheroid, paraboloid, prolate spheroid, pseudosphere, sphere, spheroid, and torus (and its generalization, the toroid).

The area element of the surface of revolution obtained by rotating the curve from to about the x-axis is

			(1)
			(2)

so the surface area is

			(3)
			(4)

(Apostol 1969, p. 286; Kaplan 1992, p. 251; Anton 1999, p. 380). If the curve is instead specified parametrically by , the surface area obtained by rotating the curve about the x-axis for if in this interval is given by

(5)

Similarly, the area of the surface of revolution obtained by rotating the curve from to about the y-axis is given by

			(6)
			(7)

(Anton 1999, p. 380). If the curve is instead specified parametrically by , the surface area obtained by rotating the curve about the y-axis for if in this interval is given by

(8)

The following table gives the lateral surface areas for some common surfaces of revolution where denotes the radius (of a cone, cylinder, sphere, or zone), and the inner and outer radii of a conical frustum, the height, the eccentricity of a spheroid, and and the equatorial and polar radii (for a spheroid) or the radius of a circular cross-section and rotational radius (for a torus).

surface
cone
conical frustum
cylinder
oblate spheroid
prolate spheroid
sphere
torus
zone

The standard parameterization of a surface of revolution is given by

			(9)
			(10)
			(11)

For a curve so parameterized, the first fundamental form has

			(12)
			(13)
			(14)

Wherever and are nonzero, then the surface is regular and the second fundamental form has

			(15)
			(16)
			(17)

Furthermore, the unit normal vector is

(18)

and the principal curvatures are

			(19)
			(20)

The Gaussian and mean curvatures are

			(21)
			(22)

(Gray 1997).

Pappus's centroid theorem gives the volume of a solid of revolution as the cross-sectional area times the distance traveled by the geometric centroid as it is rotated.