Final answer:
To find the fluctuation of the shape of a body, various principles can be applied depending on context, such as Brownian motion for small particles in a fluid. Geometrically, shapes with the same volume may have different surface areas depending on their form. The surface area of a sphere is calculated by the formula 4πr² and its volume by (4/3)πr³.
Step-by-step explanation:
To find the fluctuation of the shape of a body, there isn't one formula that can be universally applied, as it widely depends on the context in which the shape fluctuation is studied. When discussing shape fluctuations related to bodies immersed in a fluid, for example, one might refer to principles observed in Brownian motion, where the fluctuations are caused by the collisions of fluid molecules with the body. In this context, fluctuations can be significant for small particles and are roughly proportional to the inverse square root of the number of collisions.
Concerning geometrical shape changes while keeping volume constant, we often find that departures from a spherical shape lead to increases in surface area. This phenomenon is captured by the Laplace law in fluid mechanics, which says that for surfaces with equal volume, shapes like filaments or sheets will have a higher surface area than a sphere.
For calculating the surface area of a geometric body, such as a sphere, the formula is 4πr², and for its volume, the formula is (4/3)πr³. The fluctuation in shape typically involves a more complex set of calculations often involving differential equations and principles of dynamics.