# hexagon

• Dec 1st 2013, 07:38 PM
vegasgunner
hexagon
https://data.artofproblemsolving.com...e10c907669.png

• Dec 1st 2013, 07:45 PM
LimpSpider
Re: hexagon
Do you want us to find and explain the solution? Or just explain the solution without giving the solution? Where has your working taken you so far?
• Dec 1st 2013, 07:54 PM
vegasgunner
Re: hexagon
I just would like and explanation please
• Dec 1st 2013, 08:13 PM
ibdutt
Re: hexagon
Attachment 29839 with this much you should be able to complete
• Dec 2nd 2013, 04:13 PM
vegasgunner
Re: hexagon
Quote:

Originally Posted by ibdutt
Attachment 29839 with this much you should be able to complete

thank you I have solved it to be 4/3
• Dec 2nd 2013, 04:58 PM
HallsofIvy
Re: hexagon
Good! I presume that you realized from idbutt's picture that the if we take the distance from the center to a vertex of the outer hexagon to be 1, then the distance from the center to the middle of a side of that hexagon is $\displaystyle \sqrt{3}/2$. And that is where the circle touches the hexagon so is a vertex of the inner hexagon. Since those are "equivalent" distances, the ratio of the areas is the square of the ratio of those distances: $\displaystyle \left(\frac{1}{\sqrt{3}/2}\right)^2= \frac{1}{3/4}= \frac{4}{3}$.