Circles within a Circle: how many cups can fit in steamer?

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uberathlete

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Hi everyone. Here's my question: How many 1 inch diameter circles would fit into a 9 inch diameter circle?

This isn't for homework or anything like that. I'm just trying to figure out how many circular tartlet cups would fit into a bamboo steamer (I need to know because I'm gonna purchase some tartlet cups online). :mrgreen: The answer would be great but how to get the answer would be even better so that I can vary the diameters. I apologize because I have absolutely no idea how to solve this problem. Any help would be much appreciated. Thanks!
 
Re: Circles within a Circle

It may depend on how you pack them. The packing problem is an excellent exploration. Such an exploration often is done in 3D with balls in a box.

In any case, it is easy enough to create an upper bound simply by dividing the area.

9" Diameter Circle Area = \(\displaystyle \pi (9"/2)^{2}\;=\;63.617\;in^{2}\)

1" Diameter Circle Area = \(\displaystyle \pi (1"/2)^{2}\;=\;0.785\;in^{2}\)

Maximum number if you cut them up and cram them in to fill ALL the available space. 63.617/0.785 = 81.041

So, you certainly could not fit in more than 82.

The next approximation might be to calculate the wasted space for two circles sitting next to each other. It may, or may not, be obvious, but a quick chunk of geometry suggests:

1 - 0.785 = 0.215

This leads to a little algebra: \(\displaystyle \frac{63.617 - (n-1)0.215}{0.785}\;=\;n \implies n = 63.832\)

I've considered only a one-dimensional arrangement. Actually thinking about the two-dimensional arrangement could refine the estimate a little better and it may be good news or it may be bad news. Also, I looked only at straight lines. Lining them up around curves will produce different results.

If it were me, I'd shoot for 65, just to keep a couple of spares around. Do tartlet cups wear out?
What's the harm in UNDERbuying? Do they have to be packed in tightly?
Is there a cost limitation or have you sufficient funds to buy as many as you need?

Note to Students: This illustrates the beauty and utility of mathematics. It grants us the ability to answer interesting and useful questions, even if we have not encountered such a question in the past.
 
Re: Circles within a Circle

How many 1 inch diameter circles would fit into a 9 inch diameter circle?

This isn't for homework or anything like that. I'm just trying to figure out how many circular tartlet cups would fit into a bamboo steamer (I need to know because I'm gonna purchase some tartlet cups online). The answer would be great but how to get the answer would be even better so that I can vary the diameters.
Click to expand...

Hi, uberathlete,

Tkhunny has given you a straightforward approach to your problem, showing you how to bracket your solution with upper and lower bounds – in this case between 63 and 81 tartlet cups.

We can tighten this range by lowering the upper bound if we do a little research. It has been shown that the highest packing density possible for circles in a plane is about .907. See

http://www.math.sdu.edu.cn/mathency/math/c/c322.htm

What this means is that we should take the area of the steamer and multiply by .907, then divide by the area of one tartlet cup in order to estimate the maximum number possible. Always round this answer down. For example,

Steamer area = 63.617 in^2
Maximum usable area = (63.617)(.907) = 57.694 in^2
For tartlet cup diameter = 1”, area = .785 in^2
N = 57.694/.785 = 73.459
Round answer down to n = 73 (maximum number of cups)

Thus, we have bracketed our solution to between 63 and 73 cups.

Using the same approach,
Max number of 2” diameter cups is 18.
Max number of 3” diameter cups is 8. (Note: Actual answer is 7.)

Hope that helps.
 
Re: Circles within a Circle

Wow thank very much! Your replies were a great help. I'm trying to fit in as many tartlet cups/molds as I can. I guess I'll have to purchase a bit more. In the future, I'm hoping to get to the point of having an even larger steamer with many more cups such as shown here:

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Thank you once again for guidance on this matter, I appreciate it.
 
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