Expert: Sherry Wallin - 12/3/2009

Question
QUESTION: I work at a paper mill and I am trying to find a formula that will help us calculate the number of paper rolls that will sit on a trailer floor. The diameter of our rolls vary but they are always placed in a staggered pattern in the trailer. Each roll must touch the trailer wall (first the left wall - next roll touches the right wall and blocks the previous roll tight into place). So when the first roll is placed in the front left corner of the trailer the second roll touches the first roll as well as the right trailer wall. The rolls are too large to sit side by side.

Average roll is 72" diameter - trailer is 9' wide and 48' long so normally 9 rolls fit. Is there a formula I can use to re-calculate the numbers of rolls that will fit when the roll diameter changes ??

ANSWER: What is the length of each roll? How much overlap is there with each pair of rolls? What you have described is an isosceles triangle problem so it is necessary to have the length of the rolls. If I know the length of the rolls and the amount of overlap I can figure the angels out and thus the area each roll takes up.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: The rolls are standing up so the 72" diameter is what takes up the floor space.  

Answer
OK I think I get the picture now. Essentially if the rolls are 72" then you will fit 72" + a 36" chord (that is a line that goes from one end of a circle to the other and a diameter is the longest chord in a circle). In this case since the rolls are 72" we can only fit that part of the circle that measures 36" across. If your rolls will be the same but unknown I can give you a formula that will tell you how many will fit. If the diameter is arbitrary for each roll, that cannot be figured out exactly. (that is, without calculating for each roll). Let me know if this is the case and I will develop a formula for you.

Math Prof