How many tubes in a cylinder - circles in a circle

[Robster59]

Looking for a little help. I am trying to work out how many tubes of 13mm diameter will fit inside a canister of

1. 71mm
2. 102mm

I've looked online and found one website (Engineering Toolbox) but that didn't provide me with a result of any kind.
I then found this (Planet Calc) and that gave me 21 for a 71mm diameter and 48 for a 102mm diameter canister, which seems high. Although it is confirmed by this website (EasyCalculation)

I'm just looking to sanity check with anyone else out there who can confirm this makes sense.

Thanks in advance.

 

[ExRabbit]

Looking for a little help. I am trying to work out how many tubes of 13mm diameter will fit inside a canister of

1. 71mm
2. 102mm

I've looked online and found one website (Engineering Toolbox) but that didn't provide me with a result of any kind.
I then found this (Planet Calc) and that gave me 21 for a 71mm diameter and 48 for a 102mm diameter canister, which seems high. Although it is confirmed by this website (EasyCalculation)

I'm just looking to sanity check with anyone else out there who can confirm this makes sense.

Thanks in advance.

Why do you want to know this?

 

[rulie]

I found 21 and 45 from those websites.

 

[srixon 1]

Are you making a steam engine.

 

[rudebhoy]

Are you trying to work out how many Pringles you get in the big and the little tubes?

 

[Slab]

That's one tough pub quiz, hope you win

 

[Robster59]

Why do you want to know this?

Are you making a steam engine.

The reason is to find out how many tubes for IVF straws will fit into a canister in a cryogenic storage dewar.
I bet you're glad you asked.

 

[YandaB]

The reason is to find out how many tubes for IVF straws will fit into a canister in a cryogenic storage dewar.
I bet you're glad you asked.

And does the size of the tube change depending on temperature? Do the sizes of the large and small tubes change at the same rate?

 

[backwoodsman]

I found this an intriguing question - so after making my head hurt for a bit, I browsed ..

And found a calculator ( https://jcmiller11.github.io/circlepacking/ ) that suggests 22 13mm tubes will fit inside a 71mm canister (and 48 within the 102mm canister

 

[Fade and Die]

The reason is to find out how many tubes for IVF straws will fit into a canister in a cryogenic storage dewar.
I bet you're glad you asked.

That was going to be my guess actually

 

[Robster59]

That was going to be my guess actually

A man in the know.

 

[Alan Clifford]

Get a piece of paper and draw the big circle on it. Cut out the small circle from another piece of paper. Put the small circle in the big circle and draw around it. Move the small circle and draw around it. Move the small circle ...

 

[backwoodsman]

Get a piece of paper and draw the big circle on it. Cut out the small circle from another piece of paper. Put the small circle in the big circle and draw around it. Move the small circle and draw around it. Move the small circle ...

Doesnt actually work that way though, does it? Unless you happen to move the small circle to exactly the right spot each time, you probably won't get the correct answer.

 

[jim8flog]

I was going to do the maths but as somebody has already done it and made a chart ...............

 

[clubchamp98]

Do the tubes have internal or external caps?
The amount you would get in is less if it’s external.
The size of the cap is what needs to be measured .!

 

[Robster59]

Doesnt actually work that way though, does it? Unless you happen to move the small circle to exactly the right spot each time, you probably won't get the correct answer.

Well, I could have tried that, but I was trying for an easier solution. Doing a favour for a customer.

 

[Robster59]

Do the tubes have internal or external caps?
The amount you would get in is less if it’s external.
The size of the cap is what needs to be measured .!

It's straws on canes inside the tube. The customer showed it to me on Teams so the 13mm is the max dimension of each tube that would be put inside the canister.

 

[Deleted member 29109]

Chat GPT says 19 for the 71mm and 37 for the 102.

To determine how many 13 mm diameter circles can fit inside a 71 mm diameter circle using hexagonal packing, we need to follow a systematic approach:

1. **Calculate the radius of both circles**:
- Radius of the large circle, \( R \) = 71 mm / 2 = 35.5 mm
- Radius of the small circle, \( r \) = 13 mm / 2 = 6.5 mm

2. **Arrange circles in layers**:
- The distance from the center of one small circle to the center of an adjacent small circle is \( 2r \) = 13 mm.
- Circles are arranged in a hexagonal pattern around a central circle.

3. **Determine the number of layers**:
- The distance from the center of the large circle to the center of circles in the \( n \)-th layer is approximately \( n \times 2r \).
- Calculate the maximum number of layers (\( n \)) that fit within the radius \( R \):
\[
R = n \times 2r \implies 35.5 = n \times 13 \implies n \approx 2.73
\]
- So, there can be 2 complete layers around the central circle.

4. **Count the circles**:
- **Layer 0**: 1 circle (center)
- **Layer 1**: 6 circles (1st layer around the center)
- **Layer 2**: 12 circles (2nd layer around the center)

So, let's verify the arrangement and total count of circles:

1. **Layer 0** (center): 1 circle
2. **Layer 1** (around the center): 6 circles
3. **Layer 2** (around the first layer): 12 circles

Adding them together:
\[
1 + 6 + 12 = 19 \text{ circles}
\]

Thus, 19 circles of 13 mm diameter can fit inside a 71 mm diameter circle using efficient hexagonal packing.

 

[Robster59]

Chat GPT says 19 for the 71mm and 37 for the 102.

To determine how many 13 mm diameter circles can fit inside a 71 mm diameter circle using hexagonal packing, we need to follow a systematic approach:

1. **Calculate the radius of both circles**:
- Radius of the large circle, \( R \) = 71 mm / 2 = 35.5 mm
- Radius of the small circle, \( r \) = 13 mm / 2 = 6.5 mm

2. **Arrange circles in layers**:
- The distance from the center of one small circle to the center of an adjacent small circle is \( 2r \) = 13 mm.
- Circles are arranged in a hexagonal pattern around a central circle.

3. **Determine the number of layers**:
- The distance from the center of the large circle to the center of circles in the \( n \)-th layer is approximately \( n \times 2r \).
- Calculate the maximum number of layers (\( n \)) that fit within the radius \( R \):
\[
R = n \times 2r \implies 35.5 = n \times 13 \implies n \approx 2.73
\]
- So, there can be 2 complete layers around the central circle.

4. **Count the circles**:
- **Layer 0**: 1 circle (center)
- **Layer 1**: 6 circles (1st layer around the center)
- **Layer 2**: 12 circles (2nd layer around the center)

So, let's verify the arrangement and total count of circles:

1. **Layer 0** (center): 1 circle
2. **Layer 1** (around the center): 6 circles
3. **Layer 2** (around the first layer): 12 circles

Adding them together:
\[
1 + 6 + 12 = 19 \text{ circles}
\]

Thus, 19 circles of 13 mm diameter can fit inside a 71 mm diameter circle using efficient hexagonal packing.

I never thought of ChatGPT. Thanks.

 

[Voyager EMH]

Normal hexagonal packing does not necessarily maximise circles within a circle.

The Engineering Toolbox calculator does not give the maximum. I tested it.
For radius 1 small circles within radius 5 large circle it gives 17. The answer should be 19.

I could only find online proofs for up to 20 circles within a larger circle.

Circles in Circles

I think that for the OP example, although 21 will undoubtedly fit, I think it is possible that 22 might be the maximum.
For the radius 6.5 circles inside a radius 51 circle the Engineering Toolbox calculator gives 45. This will work in practicality, but I think that 48 might be possible.
I am unable to prove either of these postulations at present.