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
Surface tension , which is the angle the tangent to the surface makes with the solid surface. Note that the angle is measured "through the liquid", as shown in the diagrams above. The diagram to the right shows two examples. Tension forces are shown for the liquid–air interface, the liquid–solid interface, and the solid–air interface. The example on the left is where the difference between the liquid–solid and solid–air surface tension, , is less than the liquid–air surface tension, , but is nevertheless positive, that is In the diagram, both the vertical and horizontal forces must cancel exactly at the contact point,