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The midpoints of a regular hexagon are connected to form another regular hexagon.  Find the ratio of the area of the smaller hexagon to the larger hexagon.

 

 Dec 21, 2020
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Let the side length  of the larger hexagon  = S

 

And we can find the side length of  the  smaller hexagon as

 

sqrt [  2 (1/2 S)^2   - 2 ( 1/4 S^2) cos (120°) ]

 

sqrt [ 2 (1/2 * S)^2  -  2 (1/4)S^2  (-1/ 2) ]  =

 

sqrt [ (1/2)S^2  +  (1/ 4) S^2  ]   =

 

(S) sqrt [  3/4 ]   

 

These are similar figures so the ratio of  their  areas  = [ scale factor ] ^2  =  [ sqrt (3) / 2 ]^2  =  3/4

 

 

cool cool cool

 Dec 21, 2020

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