you're not going to like the answer if you don't like math, especially geometry (the only math I ever go above a "C" in -- Aced it in fact!!)
step 1: find the circumference (with icing applied) of the tier in question --
for sake of simplicity -- let's say we are working with a 9" tier that with icing is now 10" (that's a 1/2 inch of icing all the way around)
so...digging out the HS math: circumference = pi*D so that's 3.14 * 10 giving a value of 31.4 inches.
now comes the fun part -- figuring out those panels
easiest way: decide on an arbitrary # of panels for each tier and then divide the circumference by the # of panels you want on a tier. (we're trading on the concept of a circle being an infinite # of sides equidistant from a central point only in this case we're deciding just how many sides that circle will have -- in essence converting it to a regular polygon)
so let's say you want 12 panels 31.4/12 = 2.616666666666667 or just over 2.5" that will be the DIAMETER of those panels -- NOT the width of material they are made of which is 1/2 the circumference for that diameter.
so we take 2.5 * 3.14 = 7.85 (total circumference) / 2 = 3.925 for the width of material that will be formed to make a half circle that is 2.5" across at the open side.
so here it all is on one formula:
(((pi*diameter of tier)/# of panels on tier)*pi)/2 = width of material needed.
and where the value arrived at just before the final division by 2 is the width diameter (open side of half-circle) of the final curved panel.
now, let's say just for grins and giggles (yes I do love geometry that much that I find it full of grins and giggles!) you want a standardized size for all the panels on the cake:
well golly gee -- that PI comes to the rescue and doesn't hit us in the face!
SIMPLE -- just make all the panels out of a piece of material 3.14159........ wide and shape them to have a diameter of 2" (or 6.28318 and diameter of 4, etc.).
the problem with the cheat -- you'll never get an exact fit to the cake -- will nearly always be a scoch tight or loose fit.
clear as mud (cake) right?
just had another thought:
if you have a set of flower formers -- those could be used to both measure and make the plaques. THO' they will not be full half-rounds then as flower formers are only about 1/4 - 1/3 of circle.
which then gets mind going to:
using tp or paper towel roll split in half to measure and make
these methods will be less precise and result in less than perfect fit -- but close enough can be good enough
I am NOT a math whiz... far from it! I like Doug's 2nd option and that's basically what I'd do...
I'd determine what I was going to use as my "mold" (flower former, cardboard tube, etc) and then use it to measure up against the cake, leaving a slight space between where you measure... which would make allowance for the plaque being slightly larger than the former. As Doug mentioned, they may not fit perfectly once everything is put in place, but you can adjust them slightly and of course, that's what the back is for! Good luck - would love to see pics of your finished cake!
Amazing cake!! and amazing Doug! This is where I just yell 'Honey can you figure this out!!
Seriously I have chocolate moulds which are reasonably similar or you could fold acetate into the shapes and go the chocolate route!
All I can say after reading this is WOW!! My head hurts, lol. Good Luck!