Do I have enough iMac boxes to make a full circle? My work has a bunch of iMac boxes and because of their slightly wedged shape we are curious how many it would take to make a complete circle.
We already did some calculations and also laid enough out to make 1/4 of a circle so we know how many it would take, but I'm curious to see how others would approach this problem mathematically and see if you came up with the same answer.
Our stock consists of 12 of the 27" iMac boxes and 16 of the 21.5" iMac boxes.
The 21.5" box has the following dimensions

top: 5"
height: 21.25"
bottom: 8.75"

and the 27" box is

top: 5.75"
height: 23.75"
bottom: 9.5"

21.5" iMac box picture for reference


Trying to avoid trigonometric functions: The outer circle must be longer than the inner circle by $2\pi$ times the height, so compute $$\frac{2\pi\text{height}}{(\text{bottom}-\text{top})} $$
as a good approximation.
