I’m 70, so no, this isn’t homework. I’m usually pretty good at this stuff, but this has me stumped. I’m planning to make mince tarts this week, which requires pie dough cut into small circles, say about 3" in diameter. Rather than guess how many pie crusts I need to buy, I’d like to be able to come up with an estimate.
So the question is, how many 3" circles can I get from a 9" pie crust? If someone can provide a formula for figuring this out, I’d appreciate it.
You’ll be able to get three across at the center. You’ll be able to get three down at the center. That makes five, and I’m not thinking there’s any more efficient use of the space, since any way you rotate it will lead to a similar pattern.
You could instead get two across as soon as there’s an arc with a length of 6 inches, but I think that’d lead to a total of four, not five.
(Radius of larger)[sup]2[/sup] ÷ (radius of smaller)[sup]2[/sup] = number of smaller that can be made from larger … both factors have a π term but I’ve taken the liberty to cancel those out for you …
The answer to the example is 9[sup]2[/sup] ÷ 3[sup]2[/sup] = 9 …
The amount of dough in a 9" diameter (4.5" radius) circular pie crust (pi * 4.5²) is 9 times as much as the amount of dough in a 3" pie crust (pi * 1.5²).
On preview, I see that watchwolf49 has already calculated this. He used diameters instead of radii in his calculation, but it doesn’t matter if all you care about is the ratio.
If you roll out the remaining scraps then you can make 9[sup]2[/sup] / 3[sup]2[/sup] = 9, of course. (The pis cancel out; hopefully the pies won’t.)
Edit: beaten to it.
This is really embarrassing, but then it’s only just 6 a.m. and I’ve been up for two hours without coffee. The above seems logical. But what if I started at the top and then worked around the perimeter with the small circles touching each other? Do I gain anything, or do I end up the same?
If you can salvage the scraps and roll them out again to the original thickness, you can get exactly 9 tarts from one pie crust (nine squared divided by three squared). If the scraps are waste and can’t be reused, you can get 7 three-inch circles from one nine-inch crust. (cite)
It just occurred to me that we should have asked what you meant when you said a nine-inch crust. Is it a crust with a known diameter of nine inches or a crust designed to fit in a 9-inch pie pan? If the latter, it will have to be bigger than 9 inches in diameter, in order to fit up the sides with some left over for edges. The ones I’ve bought in the past were probably at least 12 inches in diameter, maybe 13 or a little more. Assuming 12 inches you should be able to get 16 three-inch circles if you reuse the scraps.