• Ramani

MemberOct 25, 2013

## Rolled up Yoga Mat(h)

This is really an elementary problem in geometry and algebra. It is not puzzling nor are there any twists. The result is interesting.

I came across this intriguing problem a couple of days back. On my return from my recent trip to USA I found that my brother in law from US, who was on a reverse visit to India, had left behind one of those Yoga Mats of compressed foam. I normally use a simple folded blanket. The mat was kept tightly rolled in its card board carton. Presumably this is done to reduce the carton size.

I had to go through some non yoga contortions to straighten out the mat prior to use. Because it was tightly rolled it tended to curl up doing yoga of its own.

While doing yoga I got to considering what is the relationship between the ID (that is, the starting diameter) of the rolled mat and its final OD.

The mat is 200 cms long and 0.5 cm thick. Given this data it is fairly elementary algebra and geometry to work out the analytical relationship between these parameters.
Finally it ended up as a two column excel file. The first column is the starting dia of the roll (the central hole as it were) and the second is the (calculated) outer dia of the roll.

The results were quite unexpected and revealing.

Those curious can have a shot at this.
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Replies
• MemberOct 26, 2013

Raison de etre for the exercise:
How much effect will making the roll have a larger central opening, which makes it easier to unroll, have on the OD of the roll and hence the carton cost?
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• MemberOct 29, 2013

One can use calculus, numerical methods or even more complex ways. I just assumed that the rolled up mat was a hollow cylinder with ID as the variable. The cross sectional area of this should be the area of the mat's edge. In this case 200 x 0.5 = 100 sq.cm.
Pi x (D^2 - d^2)/4 = 100
D = Sqrt((400/Pi) - d^2)
Solving this for various values of d is a no brainer.

I found that for d varying from 0 to 5 cm the OD of the roll does not change by more than 1 cm.
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