Question-219721

Question Number 219721 by Spillover last updated on 01/May/25

Answered by mr W last updated on 01/May/25

Commented by mr W last updated on 01/May/25

r=radius of sectors a=side length of hexagon a=[(1−((√3)/2))×(2/( (√3)))+1]r=((2r)/( (√3))) ((sectors)/(hexagon))=(((πr^2 )/6)/(((√3) a^2 )/4))=((2π)/(3(√3)))×(((√3)/2))^2 =(π/( 2(√3)))≈0.907

$${r}={radius}\:{of}\:{sectors} \\ $$$${a}={side}\:{length}\:{of}\:{hexagon} \\ $$$${a}=\left[\left(\mathrm{1}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)×\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}+\mathrm{1}\right]{r}=\frac{\mathrm{2}{r}}{\:\sqrt{\mathrm{3}}} \\ $$$$\frac{{sectors}}{{hexagon}}=\frac{\frac{\pi{r}^{\mathrm{2}} }{\mathrm{6}}}{\frac{\sqrt{\mathrm{3}}\:{a}^{\mathrm{2}} }{\mathrm{4}}}=\frac{\mathrm{2}\pi}{\mathrm{3}\sqrt{\mathrm{3}}}×\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\pi}{\:\mathrm{2}\sqrt{\mathrm{3}}}\approx\mathrm{0}.\mathrm{907} \\ $$

Commented by Spillover last updated on 01/May/25