A regular hexagon is inscribed in another regular hexagon with side lengths of 2 ft as shown. What is the ratio of the areas (smaller to larger)?
Choose a vertex on the outer hexagon; call this point "A".
Choose the nearest two vertices on the inner hexagon; call these points "B" and "C".
In triangle(ABC), AB = 1 and AC = 1.
Knowing these two, you can find B] it is aqrt(3) [I used the Law of Cosines; you can drop of perpendicular from A to BC and use either triangle.]
The ratio of the areas of two similar polygons is equal to the ratio of the squares of corresponding sides.
Therefore, the ratio of the area of the outer hexagon to the inner hexagon is: 22 / sqrt(3)2 = 4/3.