Paper bag problem

Key takeaways

Paper bag problem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Paper bag problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Paper bag problem from memory before moving on to harder problems.

Reference excerpt

In geometry, the paper bag problem or teabag problem is to calculate the maximum possible inflated volume of an initially flat sealed rectangular bag which has the same shape as a cushion or pillow, made out of two pieces of material which can bend but not stretch. According to Anthony C. Robin, an approximate formula for the capacity of a sealed expanded bag is:

V = w 3 ( h / ( π w ) − 0.142 ( 1 − 10 ( − h / w ) ) ) , {\displaystyle V=w^{3}\left(h/\left(\pi w\right)-0.142\left(1-10^{\left(-h/w\right)}\right)\right),}

where w is the width of the bag (the shorter dimension), h is the height (the longer dimension), and V is the maximum volume. The approximation ignores the crimping round the equator of the bag. A very rough approximation to the capacity of a bag that is open at one edge is:

V = w 3 ( h / ( π w ) − 0.071 ( 1 − 10 ( − 2 h / w ) ) ) {\displaystyle V=w^{3}\left(h/\left(\pi w\right)-0.071\left(1-10^{\left(-2h/w\right)}\right)\right)}

(This latter formula assumes that the corners at the bottom of the bag are linked by a single edge, and that the base of the bag is not a more complex shape such as a lens).

The square teabag

For the special case where the bag is sealed on all edges and is square with unit sides, h = w = 1, the first formula estimates a volume of roughly

V = 1 π − 0.142 ⋅ 0.9 {\displaystyle V={\frac {1}{\pi }}-0.142\cdot 0.9}

or roughly 0.19. According to Andrew Kepert, a lecturer in mathematics at the University of Newcastle, Australia, an upper bound for this version of the teabag problem is 0.217+, and he has made a construction that appears to give a volume of 0.2055+. Robin also found a more complicated formula for the general paper bag, which gives 0.2017, below the bounds given by Kepert (i.e., 0.2055+ ≤ maximum volume ≤ 0.217+).

See also Biscornu, a shape formed by attaching two squares in a different way, with the corner of one at the midpoint of the other Mylar balloon (geometry)

Notes

References Robin, Anthony C (2004). "Paper Bag Problem". Mathematics Today. June. Institute of Mathematics and its Applications: 104–107. ISSN 1361-2042. Weisstein, Eric W. "Paper Bag". MathWorld. Archived from the original on 2011-06-29.

External links The original statement of the teabag problem Andrew Kepert's work on the teabag problem (mirror) Curved folds for the teabag problem A numerical approach to the teabag problem by Andreas Gammel Weisstein, Eric W. "Paper Bag Surface". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Paper bag problem

Start with the simplest possible case. Write down what Paper bag problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Paper bag problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Paper bag problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Paper bag problem

In research: Paper bag problem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Paper bag problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Paper bag problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometric shapes, Mathematical optimization, so understanding it makes those chapters shorter.
In everyday life: Look for Paper bag problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Paper bag problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Paper bag problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Paper bag problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Paper bag problem in simple terms?

In geometry, the paper bag problem or teabag problem is to calculate the maximum possible inflated volume of an initially flat sealed rectangular bag which has the same shape as a cushion or pillow, made out of two pieces of material which can bend but not stretch. According to Anthony C.

Why does Paper bag problem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Paper bag problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Paper bag problem.