Equilateral Triangle within a Rectangle 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:

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.
 A: A now-deleted previous answer had seven triangles. With the help of GeoGebra to draw an accurate figure, I managed eight.

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.
