Triangle + rectangle = Trirectangle?

Geometry Level pending

One day Mohamed got himself interesting about the equilateral triangles and all the symmetric stuffs. He had a piece of paper in one of his hand in the shape of a rectangle and he start to think about how much equilateral triangles he could put inside his paper, with the condition that all the three vertices of the triangle coincide with the board of the paper - like a regular person.

Decided to answer this question, he created his two first math formula which could predict how much paper is required to put " \(n\)" numbers of equilateral triangles with sides "\(s\)" on the paper. And even better, his formula is based on the economic strategy of the paper (with the least waste of paper) .

The first one, tells about the width of the paper based on the proportions of the triangle, the second one relates the number of triangles with the length of the paper. Which are these "magical formulas"?

\(\textbf{Details and assumptions:}\)

*This is an original problem (I think...), and my first problem here. In the case of ambiguity, just let me know to fix it.

Enjoy it :)


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