Problem:
Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of △ABC?
Answer Choices:
A. 23
B. 33
C. 1+32
D. 2+23
E. 3+23
Solution:
Label points E and F as shown in the figure, and let D be the midpoint of BE. Because △BFD is a 30−60−90∘ triangle with hypotenuse 1, the length of BD is 23, and therefore BC=23. It follows that the area of △ABC is 43⋅(23)2=(B)33.
OR
Notice that AE=3 since AE is composed of a hexagon side (length 1) and the longest diagonal of a hexagon (length 2). Triangle ABE is 30−60−90∘, so BE=33=3. The area of △ABC is AE⋅BE=(B)33.
