Problem:
Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of â–³ABC?
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=3​3​=3​. The area of △ABC is AE⋅BE=(B)33​​.
