The polygon has area 7/2 * s^2 * cot( pi /14), where s is the side length.
The rectangle has height s * cot( pi /14), so area s^2 * cot(pi/14)
We can get both these from the formula for the length of the apothem which is s / (2 * tan( pi / N ) ) for an N-sided polygon. This is half the height of the rectangle, and it is the height of each of the 14 triangles with base s that make up the polygon.
So, the ratio is 7/2 : 1, that is, 7 : 2.
That's great, but, just for educational purposes of our readers, there is a geometric method without the need for the trigonometry formulas.
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But we invented trigonometry (and algebra) so we didn't have to be clever about geometry any more! :)
I actually started out with a triangle dissection but I couldn't figure out the trick showing that you could pack exactly four triangles into the rectangle.
Now that I say that, it's probably something like using "all triangles with the same height and base have the same area" but I tend not to think in terms of non-rigid transformations.
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