Regular hexagons are very special—divide the hexagon with three diagonals (running through the center) and you will get six equilateral triangles. Why are the triangles equilateral? Since the sum of the angles in any polygon is (n – 2)(180), the sum for a hexagon is 720. Divide by 6 to get that each angle is 120. When you divide the hexagon into triangles, you split each 120 to make two 60 degree angles for each triangle. Any triangle that has two angles of 60 must have a third angle of 60 as well, since triangles always sum to 180.
Since the triangles are equilateral, we know that all of them have all sides equal to 2√3. For each triangle, we know that the height will always equal half the side times √3. (This is a good fact to simply memorize; however, had you not memorized this, you could divide each equilateral into two 30–60–90 triangles and use the side ratios of 1 : √3 : 2 for a 30–60–90 triangle.)
Therefore, the height of each triangle is simply √3 * √3 = 3.
Since the area of a triangle is [(1/2)bh], the area of each equilateral triangle is: (1/2)*(2√3)*3 = 3√3.
Since there are six such equilateral triangles in each hexagon, the area of each hexagon is 18.
Since there are six hexagons in the honeycomb, the total area of the figure is 108√3 = 187.061487....
The correct answer is A.
This week, we've got another Data Interpretation challenge:
Go to it! My own answer will eventually appear in the comments.