Problem:
Equilateral △ABC has side length 1, and squares ABDE,BCHI, and CAFG lie outside the triangle. What is the area of hexagon DEFGHI?
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Answer Choices:
A. 412+33
B. 29
C. 3+3
D. 26+33
E. 6
Solution:
The three squares each have area 1, and △ABC has area 43. Note that ∠EAF=360∘−60∘−2⋅90∘=120∘. Thus the altitude from A in isosceles △EAF partitions the triangle into two 30−60−90∘ right triangles, each with hypotenuse 1. It follows that △EAF has base EF=3 and altitude 21, so its area is 43. Similarly, triangles GCH and DBI each have area 43. Therefore the area of hexagon DEFGHI is 3⋅43+3⋅1+43=(C)3+3.
