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 \(h\): \[r =\sqrt{\frac{V_0}{\pi h}}.\] 

1.1 Linearly increasing height \(h\)

If we set the height \(h\) to be an integral multiple of the smallest height \(h_1\), then we may define the height of the \(k^{th}\) cylinder as \(h_k = k h_1\), where \(k = \{1, 2, \ldots, n\}\) and \(n\) is the number of the cylinders. The corresponding radius \(r_k\) of the  \(k^{th}\) cylinder is \[r_k =\sqrt{\frac{V_0}{\pi kh_1}}.\] Notice that the radius \(r_k\) is inversely proportional to \(\sqrt{k}\).

1.2 Linearly increasing radius \(r\)

If we set the radius \(r\) to be an integral multiple of the smallest radius \(r_1\), then we may define the radius of the (k^{th}\) cylinder as \(r_k = k r_1\), where \(k = \{1, 2, \ldots, n\}\) and \(n\) is the number of the cylinders. The corresponding height \(h_k\) of the  \(k^{th}\) cylinder is \[h_k =\frac{V_0}{\pi k^2r_1^2}.\] Notice that the height \(h_k\) is inversely proportional to \(k^2\).

2. Constant radius \(r\)

If the radius \(r=r_0\) is constant, then the volume \(V\) of the cylinder is inversely proportional to the height \(h\): \[V =\pi r_0^2 h.\]  

2.1 Linearly increasing height \(h\)

If we set the height \(h\) to be an integral multiple of the smallest height \(h_1\), then we may define the height of the \(k^{th}\) cylinder as \(h_k = k h_1\), where \(k = \{1, 2, \ldots, n\}\) and \(n\) is the number of the cylinders. The corresponding volume \(V_k\) of the  \(k^{th}\) cylinder is \[V_k =\pi r_0^2kh_1.\] Notice that the volume \(V_k\) is inversely proportional to \(k\).

2.2 Linearly increasing volume \(V\)

If we set the volume \(V\) to be an integral multiple of the smallest volume \(V_1\), then we may define the volume of the \(k^{th}\) cylinder as \(h_k = k h_1\), where \(k = \{1, 2, \ldots, n\}\) and \(n\) is the number of the cylinders. The corresponding height \(h_k\) of the  \(k^{th}\) cylinder is \[h_k =kV_1/\pi r_0^2.\] Notice that the height \(h_k\) is proportional to \(k\).

3. Constant height \(h\)

If the height \(h=h_0\) is constant, then the volume \(V\) of the cylinder is inversely proportional to the square of the radius \(r\): \[V =\pi r^2 h_0.\]  

3.1 Linearly increasing radius \(r\)

If we set the radius \(r\) to be an integral multiple of the smallest radius \(r_1\), then we may define the radius of the (k^{th}\) cylinder as \(r_k = k r_1\), where \(k = \{1, 2, \ldots, n\}\) and \(n\) is the number of the cylinders. The corresponding volume \(V_k\) of the  \(k^{th}\) cylinder is \[V_k =\pi k^2r_1^2 h_0.\] Notice that the volume \(V_k\) is inversely proportional to \(k^2\).

3.2 Linearly increasing volume \(V\)

If we set the volume \(V\) to be an integral multiple of the smallest volume \(V_1\), then we may define the volume of the (k^{th}\) cylinder as \(V_k = k V_1\), where \(k = \{1, 2, \ldots, n\}\) and \(n\) is the number of the cylinders. The corresponding radius \(r_k\) of the  \(k^{th}\) cylinder is \[r_k =\sqrt{kV_1/\pi h_0}.\] Notice that the volume \(V_k\) is proportional to \(\sqrt{k}\).

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