The perimeter

The perimeter of equilateral △PQR is 12. The perimeter of regular hexagon STUVWX is also 12. What is the ratio of the area of △PQR to the area of STUVWX?

Result

r =  2:3

Solution:

$p_{1}=12 \ \\ a_{1}=p_{1}/3=12/3=4 \ \\ \ \\ p_{2}=12 \ \\ a_{2}=p_{2}/6=12/6=2 \ \\ \ \\ S ... k a^2 \ \\ k=\sqrt{ 3 }/4 \doteq 0.433 \ \\ \ \\ S_{1}=a_{1}^2=4^2=16 \ \\ S_{2}=6 \cdot \ a_{2}^2=6 \cdot \ 2^2=24 \ \\ \ \\ r=S_{1}/S_{2}=16/24 \doteq \dfrac{ 2 }{ 3 } \doteq 0.6667≈ 0.6667 \doteq 2:3$

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