Platonic solid to that of a sphere of the same size (i.e. either the same surface area or the same volume.) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most. Every polyhedron has a dual (or "polar") polyhedron with faces and vertices interchanged. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs. If a polyhedron has Schläfli symbol {"p", "q"}, then its dual has the symbol {"q", "p"}. Indeed, every combinatorial property of one
Volume as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied. The volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m s]). In calculus, a branch of mathematics, the volume of a region "D" in R is given by a triple integral of the constant function formula_1 and is usually written as: The volume integral in cylindrical coordinates is and the volume integral