This is my equation for hollow cylinder volume:
$$\text{hollow cylinder volume}= 2 \cdot \pi \cdot (r_2 - r_1) \cdot \text{thk} \cdot ((r_2 + r_1) \cdot 0.5)$$
Where:
- $r_2 =$ outer radius
- $r_1 =$ inner radius
- thk $=$ thickness
- $\pi =$ the mathematical constant $\pi$
Here are the steps I've taken for simplification:
change in $r= (r_2 - r_1)$
average $r= (r_2 + r_1) \cdot 0.5$
Equation comes out to be:
$$\text{hollow cylinder volume} = 2 \cdot \pi \cdot \text{change in }r \cdot \text{average }r$$
My questions are:
- Why would the volume of a cylinder by multiplied by $2 \pi$?
- Why is average $r$ multiplied by change in $r$? (As opposed to multiplying average $r$ by average $r$ or change in $r$ by change in $r$).
The volume of a hollow cylinder $H$ depends on the inner radius $r_i$, the outer radius $r_o$, and the height $h$ of $H$. You have called the height "thickness", but forgotten about it later on.
Such an $H$ is a prism, namely the cartesian product of a plane annulus $A$ with an interval of length $h$. From basic measure principles it then follows that ${\rm vol}(H)$ is the product of ${\rm area}(A)=\pi(r_o^2-r_i^2)$ with $h$, i.e., $${\rm vol}(H)=\pi(r_o^2-r_i^2)h\ .$$ Of course you can write this formula in various ways, e.g., also in the form $${\rm vol}(H)=2\pi(r_o-r_i)\>h\>{r_i+r_o\over2}\ .\tag{1}$$ But there is no such thing that "${\rm vol}(H)$ is multiplied by $2\pi\,$". For an intuitive interpretation of $(1)$ one can argue as follows:
Partition $H$ into cylindrical shells of thickness $\Delta r$. The volume of such a shell of radius $r$ then is $\>\approx 2\pi r\>\Delta r\>h\>$, namely circumference $2\pi r$ of the shell times thickness $\Delta r$ times height $h$. This little volume is in particular a linear function of $r$. This implies that the sum of these little volumes is $2\pi$ times the sum $r_o-r_i$ of all $\Delta r$ times $h$ times the average value ${r_i+r_o\over2}$ of $r$. This brings us to $(1)$.