Montessori knobbed cylinders: the mathematics of radius, height, and volume
Volume of a Cylinder The volume \(V\) of a cylinder with radius \(r\), height \(h\) is given by \[V = \pi r^2 h.\] Notice that we have three parameters. If we hold one of them to be constant, then we can obtain the relationship between the other two parameters, especially if one of these two parameters is uniformly or linearly increasing, i.e., a linear function of a set of integers \(k =\{0, 1, 2, \ldots, n\}\), where \(n\) is an arbitrary integer ideally greater than or equal to 3. Let us consider three cases: constant volume, constant radius, and constant height. Each of these cases will have two sub-cases, with each sub-case dictating the design of a row of Montessori knobbed cylinders. Thus, in principle, you can design a total of 6 rows of knobbed cylinders, with each row containing \(n\) cylinders. 1. Constant volume \(V\) If the volume \(V=V_0\) is constant, then the radius \(r\) of the cylinder is inversely proportional to the square root of the height \(h\): \[r =\sqrt{\fra