# Interesting Math question

##### New member
So, I have a math question that I was wondering about, but I don’t know how to do it. The question is:

If a 55 gallon water drum has a diameter of 23 1/4 inch and a height of 34 1/4 inch, how much space would it take to store 100 of them tightly packed together into a square?

#### Otis

##### Senior Member
Hello. If I understand the question correctly, all 100 drums are on the ground (so both their volume and their height are irrelevant), and you need to find the area (in square feet, maybe) of the enclosing square on the ground.

If so, then consider a simpler version, first. If you had only four drums and arranged them in two equal rows (i.e., two drums in front and two drums in back), how big would the enclosing square be? Draw the four touching circles, on a piece of paper. Do you see how the diameters of two circles next to each other relate to the side length of the enclosing square?

If you understand that, can you extend the idea to 100 circles?

PS: Be sure to check out the forum's submission guidelines (here's a summary). We'd like to see what you've tried or thought about, so far. Thanks!

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#### Denis

##### Senior Member
Further to Otis' advice, "circle packing" may be involved;

#### Subhotosh Khan

##### Super Moderator
Staff member
Since the OP has 100 drums and a square area - "hex" packing may not provide any advantage (I have not checked it out though).

#### Otis

##### Senior Member
… "hex" packing may not provide any advantage …
I don't think it does. Look at the page Denis referenced (post #3). When the number of circles (n) is a square number, the arrangement is a √n × √n affair.

#### Subhotosh Khan

##### Super Moderator
Staff member
You want me to listen to Denis??!!

I am complaining to Jomo.....

#### Denis

##### Senior Member
If a 55 gallon water drum has a diameter of 23 1/4 inch and a height of 34 1/4 inch,
how much space would it take to store 100 of them tightly packed together into a square?
Well, with no "packing" assumed, then each drum can be looked at
as a 23.25 by 23.25 by 34.25 cuboid, meaning volume (space) of
18514.26525 cu. in. or 1851426.5250 cu. in. for 100
(or 1071.4275 cu. ft. or 39.6825 cu. yd.)
Game over!!

#### Otis

##### Senior Member
… each drum can be looked at as a 23.25 by 23.25 by 34.25 cuboid, meaning volume (space) … Game over!!
Yes, I had wondered about the relationship between 'space' and 'square', in the op. So, I guessed: space=area.

I'm waiting for the OP to respond.

#### Denis

##### Senior Member
Since the OP has 100 drums and a square area - "hex" packing
may not provide any advantage (I have not checked it out though).
Agree. Only ~3.12 in. as a "row reduction", but the reduction
to 9 drums every 2nd row will more than offset that gain...