## Surface Area

Surface area is the Area of a given surface. Roughly speaking, it is the amount'' of a surface, and has units of distance squares. It is commonly denoted for a surface in 3-D, or for a region of the plane (in which case it is simply called the'' Area).

If the surface is Parameterized using and , then

 (1)

where and are tangent vectors and is the Cross Product.

The surface area given by rotating the curve from to about the -axis is

 (2)

If is defined over a region , then
 (3)

where the integral is taken over the entire surface.

The following tables gives surface areas for some common Surfaces. In the first table, denotes the lateral surface, and in the second, denotes the total surface. In both tables, denotes the Radius, the height, the base Perimeter, and the Slant Height (Beyer 1987).

 Surface Cone Conical Frustum Cube Cylinder Lune (Solid) Oblate Spheroid Prolate Spheroid Pyramid Pyramidal Frustum Sphere Torus Zone

 Surface Cone Conical Frustum Cylinder

Even simple surfaces can display surprisingly counterintuitive properties. For instance, the surface of revolution of around the x-Axis for is called Gabriel's Horn, and has Finite Volume but Infinite surface Area.