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My friend wants me to answer her math challenge. She wants me to find how many equilateral triangles can be acquired from a rectangular sheet.

See image below for the dimensions:

See image below for the dimensions

I calculated the area of triangle which is 0.22436 sq meters

I also calculated the area of the sheet which is 2.9768 sq meters

Therefore, there will be 12.22 $\approx$ 12 pieces of equilateral triangle that can be acquired from that sheet.

When I revealed my answer, she told me that I was wrong and the correct answer should be 6 pieces.

Did I solved this problem wrong? I'm confident on my solution but she was serious when she told me that I got the wrong answer. Is she just making fun of me?

Thanks on your arguments.

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  • $\begingroup$ You have to take the shape of the objects into account as well. $\endgroup$ – астон вілла олоф мэллбэрг Feb 11 '18 at 9:43
  • $\begingroup$ You've calculated the possible maximum but did not take into account that the actual number depends on the shape (as pointed out by @астонвіллаолофмэллбэрг ) $\endgroup$ – zoli Feb 11 '18 at 9:45
  • $\begingroup$ does it mean that she is right? I take the shape of objects into account since I use their area. Isn't that enough? $\endgroup$ – nicy12 Feb 11 '18 at 10:01
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    $\begingroup$ @nicy12 Let's say I have a rectangular strip of paper $100$ cm long, and $0.01$ cm wide. Can you cut out a square which is $1$ cm by $1$ cm? Both shapes still have an area of $1$ square cm. $\endgroup$ – Toby Mak Feb 11 '18 at 10:05
  • $\begingroup$ That made sense to me @TobyMak. Now I want to know if which answer is right (or should I say, what is the correct answer?) $\endgroup$ – nicy12 Feb 11 '18 at 10:09
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A now-deleted previous answer had seven triangles. With the help of GeoGebra to draw an accurate figure, I managed eight.

enter image description here

As mentioned in comments to your question, your area calculation is helpful, in that you've shown that we can't possibly fit more than twelve triangles, since there simply isn't enough area available. However, the puzzle is to determine what's actually possible. (If I had a rectangle that was $1$ millimeter tall and a million kilometers wide, there'd be enough area to fill lots and lots and lots of those triangles; however, not even one triangle would actually fit inside that rectangle.)

To the question in your comment: "But do you have any tips how can I show it mathematically? The 'draw and cut' method won't be handy for bigger sizes". ... For puzzles like this ---which are types of "packing problems"--- there isn't always a formal approach; you just have to play around and see what works. (That's not to say there aren't sometime strategies, but here the problem is simple enough for playing-around.) That now-deleted answer with seven triangles seemed pretty good to me, but then I realized that I could jostle a couple of triangles to make room for one more. Maybe there's a clever way to do even better, but I'm not seeing it.

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