Surface area in the study of fractals. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in geometric measure theory. A specific example of such an extension is the Minkowski content of the surface. The below given formulas can be used to show that the surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3, as follows. Let the radius be "r" and the height be "h" (which is 2"r" for the sphere). formula_7 The discovery of this
Capillary surface and other energy potentials absent, the energy of the system is where the subscripts formula_40, formula_41, and formula_42 respectively indicate the liquid–gas, solid–gas, and solid–liquid interfaces. Note that inclusion of gravity would require consideration of the volume enclosed by the capillary surface and the solid walls. Typically the surface tension values between the solid–gas and solid–liquid interfaces are not known. This does not pose a problem; since only changes in energy are of primary interest. If the net solid area formula_43 is a constant, and the contact angle is known, it may be shown that (again, for two different fluids