Triangle + rectangle = Trirectangle?

Geometry Level 2

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 :)

w = \frac{s \times \sqrt{3} }{3} ; l = n\times (s + 1)

w = \frac{s \times \sqrt{3} }{3} ; l = s \times (n+1)

w = \frac{3s \times \sqrt{3} }{2} ; l = (n+1)\times\frac{s}{4}

All the previous answers is wrong.

w = \frac{s \times \sqrt{3} }{2} ; l = (n+1)\times\frac{s}{2}

w = \frac{3s \times \sqrt{3} }{2} ; l = n\times\frac{s}{4}

w = \frac{s \times \sqrt{3} }{2} ; l = n\times\frac{s}{2}

1 solution

Marco Antonio
Dec 16, 2015

First, we have to understand what Mohamed wants to do:

Starting with the basic. He's working with equilateral triangle (which has equal sides and equal angles) and putting them inside the rectangle. So we have to agree that he needs the area of the triangles in order to measure them in the paper. Here some simple illustration that I made:

Now comes the important part:

The height of the the triangles is the same of the width of the paper;

And the remainder paper from the triangles makes another equilateral triangle - which can be confirmed using angles properties.

Here it is how it looks like:

So here start our algebra:

First: since the height of the triangle is the same of the width of the rectangle, we already have our first formula:

$w = \frac{s \times \sqrt{3} }{2}$

Second: Considering one triangle or more in the paper ( $n$ triangles), using the efficient way, we always end up with the same are of $(n+1)$ triangles. If the area of the $(n+1)$ triangles is the same of the rectangle, we start to create our second formula:

$Area Triangle \times (n + 1) = AreaRectangle$

$\frac{s^{2}\sqrt{3}}{4} \times (n+1) = \frac{s\sqrt{3}}{2} \times leght (l)$

$l = (n+1) \times \frac{s}{2}$ .

So there it s. I hope I made it clear.

Moderator note:

Interesting problem!

To avoid ambiguity, the width of the paper should be given. For example, if we had $w = \frac{3}{4} s$ instead, then the answer would have been different.

Be careful with "efficient way". It doesn't mean "triangles are perfectly aligned", but instead just "way with least waste of paper". If the width was too short, then we will need to tilt our triangles, and hence get a different (but still) efficient way.

Interesting problem!

To avoid ambiguity, the width of the paper should be given. For example, if we had $w = \frac{3}{4} s$ instead, then the answer would have been different.

Calvin Lin Staff - 5 years, 5 months ago

Thanks for the advice. I really appreciate, and I guess the only thing I can change in the question is the idea of "efficient way" to "least waste of paper".

Marco Antonio - 5 years, 5 months ago

Triangle + rectangle = Trirectangle?

1 solution

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