The circumradius R and the inradius r of a regular hexagon:
We can find their ratio by taking one of the six equilateral triangles that make up a regular hexagon, and cutting it into two right angled triangles. Using Pythagoras' theorem then delivers the ratio r/R.
However, here is a more elegant method which avoids Pythagoras:
The area of the larger multicoloured hexagon is 3 times the area of the inner green hexagon (each colour has the area of one green hexagon).
If the inradius of the green hexagon is r, and its circumradius R, the circumradius of the multicolourd hexagon is 2r.
We know that area increases as the square of any linear dimension. The increase in area from green hexagon to multicoloured hexagon = the square of the increase in circumradius
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