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.
+128475 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
