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You are watching: How many square inches of paper needed to wrap a box

— Charles Hamilton Houston, NAACP litigation Director.

Introduction


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Figure 1: examples of Christmas GiftWrapping (Source).


My sister functions as an event planner/wedding planner. She created me an e-mail today through the following gift wrapping question.

I have to wrap 375 boxes. 16x14x6. The wrapping record is 30 inches wide. I say ns need around 1100 feet. People are telling me I just need fifty percent that.

She climate referred me to a website that offers a formula because that the lot of wrapping document for a given prismatic box. Her team was not acquainted with evaluating formulas, and also she inquiry me if I could do the calculations.

This formula was supposedly created by Sara Santos, a well-known used mathematician. However, as soon as I looked in ~ the equation (Equation 1), ns knew something was wrong. Ns am certain Sara acquired it properly – the difficulty is one of gaining it top top the web page correctly. Ns still mental a conversation I had years earlier with a typesetter who complained about setting mathematical form because it was "fussy" – he fee extra for it because it had actually to it is in done exactly right.

Here is a video that offers a good background ~ above the formula and also how the is used.For those who don"t desire the joy of computing, I include an Excel spreadsheet here to help you. No macros, simply a simple calculator.

Background

Equation as Stated

Equation 1 is claimed to provide the area of wrapping file required because that a single box.

Eq. 1
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The visibility of the equality is what perplexed me. However, ns have established what is walk on and also the equality need to not it is in there. The very first term $latex \frac12\cdot \left( d+2\cdot h+w \right)^2&s=-1$ is for rectangle-shaped boxes and the second term $latex \displaystyle 2\cdot \left( w+h \right)^2&s=-1$ is because that square box – the square crate term is a special situation of the rectangle-shaped term through d = w.

The gift pack formula does not treatment which numbers are assigned come d, w, or h. If you have actually two size that space equal, then make those d and w because you have actually a box v a square side.

Analysis

Derivation

The derivation is straightforward given two cases: (1) square box, and (2) rectangular box. Figure 3 offers my drawings of this cases. The compelled wrapping record area follows straight from these drawings.

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Figure 3(a): Square-Base Box.Figure 3(b): Rectangular-Base Box.

Now I understand that Equation 1 should not save on computer an equality. Instead, there space two separate instances covered. Really, the square-base box case is just a special situation of the rectangular-box case.

Wrapping file Constraint

The web page go not point out that the derivation provides an assumption that the wrapping file is vast enough to assistance this strategy to pack the box. To wrap a crate this way requires the the wrapping paper meet the adhering to width constraint (Equation 2).

Eq. 2$latex \displaystyle w_Wrap=\sqrt2\cdot \left( w+h \right)$square-base box
$latex \displaystyle w_Wrap=\fracw+d+2\cdot h\sqrt2$rectangular-base box

Solution to my Sister"s Problem

My sister required to know exactly how much 30-inch large wrapping file to buy. Number 4 mirrors my answer.


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Figure 4: mine Answer to my Sister"s Problem.


1100 feet of wrapping file is a the majority of paper.

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Conclusion

I love to job-related geometric problems, and my sister offered me a valuable one – everyone has to wrap packages. This problem also fits in nicely with my understand in origami and document engineering.