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=πr2h. 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,,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=V0 is constant, then the radius r of the cylinder is inversely proportional to the square root of the height h: r=V0πh. 

1.1 Linearly increasing height h

If we set the height h to be an integral multiple of the smallest height h1, then we may define the height of the kth cylinder as hk=kh1, where k={1,2,,n} and n is the number of the cylinders. The corresponding radius rk of the  kth cylinder is rk=V0πkh1. Notice that the radius rk is inversely proportional to k.

1.2 Linearly increasing radius r

If we set the radius r to be an integral multiple of the smallest radius r1, then we may define the radius of the (k^{th}\) cylinder as rk=kr1, where k={1,2,,n} and n is the number of the cylinders. The corresponding height hk of the  kth cylinder is hk=V0πk2r12. Notice that the height hk is inversely proportional to k2.

2. Constant radius r

If the radius r=r0 is constant, then the volume V of the cylinder is inversely proportional to the height h: V=πr02h.  

2.1 Linearly increasing height h

If we set the height h to be an integral multiple of the smallest height h1, then we may define the height of the kth cylinder as hk=kh1, where k={1,2,,n} and n is the number of the cylinders. The corresponding volume Vk of the  kth cylinder is Vk=πr02kh1. Notice that the volume Vk 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 V1, then we may define the volume of the kth cylinder as hk=kh1, where k={1,2,,n} and n is the number of the cylinders. The corresponding height hk of the  kth cylinder is hk=kV1/πr02. Notice that the height hk is proportional to k.

3. Constant height h

If the height h=h0 is constant, then the volume V of the cylinder is inversely proportional to the square of the radius r: V=πr2h0.  

3.1 Linearly increasing radius r

If we set the radius r to be an integral multiple of the smallest radius r1, then we may define the radius of the (k^{th}\) cylinder as rk=kr1, where k={1,2,,n} and n is the number of the cylinders. The corresponding volume Vk of the  kth cylinder is Vk=πk2r12h0. Notice that the volume Vk is inversely proportional to k2.

3.2 Linearly increasing volume V

If we set the volume V to be an integral multiple of the smallest volume V1, then we may define the volume of the (k^{th}\) cylinder as Vk=kV1, where k={1,2,,n} and n is the number of the cylinders. The corresponding radius rk of the  kth cylinder is rk=kV1/πh0. Notice that the volume Vk is proportional to k.

RELATED PRODUCTS (As an Amazon Affiliate, I earn from qualifying purchases.)

Comments