Surface area
Measure of total area of a solid object's surface.
Surface area is a measure of the total area that the surface of a solid object occupies. Its mathematical definition for curved surfaces involves infinitesimal calculus, parametric representations, and double integration, while for polyhedra it is simply the sum of the areas of flat faces.
- symbol
- A
- fundamental_property
- additivity
- key_contributors
- Henri Lebesgue and Hermann Minkowski
- related_field
- geometric measure theory
- notable_example
- Minkowski content
- historical_ratio
- sphere to cylinder surface area ratio 2:3, credited to Archimedes
Lore & Background
The rigorous definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface. A general definition should assign a positive real number to a certain class of surfaces, satisfying additivity: the area of the whole is the sum of the areas of the parts. Surface area is invariant under Euclidean motions and must agree with geometrically defined area for flat polygonal shapes.
Reader's Guide
Surface area is a fundamental geometric concept with applications across science. For piecewise smooth surfaces, a unique natural notion exists, defined via parametric surfaces and double integration of the cross product of partial derivatives. However, for irregular or rough surfaces, such as those found in fractals, assigning an area may be impossible. Extensions like the Minkowski content, studied in geometric measure theory, partially fulfill the function. A historical subtlety is the Schwarz lantern, which shows that surface area cannot be defined simply as the limit of areas of polyhedral approximations, as different choices can yield different limiting values. The ratio of surface areas of a sphere and cylinder of the same radius and height is 2:3, a discovery credited to Archimedes. In chemistry, increasing surface area generally increases the rate of a chemical reaction; in biology, it affects regulation of body temperature and digestion.
Did You Know?
- The surface area of a sphere and cylinder of the same radius and height are in the ratio 2:3, a discovery credited to Archimedes.
- Hermann Schwarz demonstrated that for a cylinder, different choices of approximating flat surfaces can lead to different limiting values of area, known as the Schwarz lantern.
- Henri Lebesgue and Hermann Minkowski sought a general definition of surface area at the turn of the twentieth century, leading to geometric measure theory.
- For very irregular surfaces, such as those with spikes spread densely, it may not be possible to assign an area at all.
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