Equilateral triangle ABC has side length of 1, and squares ABDE, BCHI, CAFG lie outside the triangle. What is the area of hexagon DEFGHI?
2 Answers
Explanation:
We will use, to find the Area of
Observe that
they all have the same Area,
Also, Area of the equilateral
Hence, The Area of the Hexagon
Area of hexagon

Area of
#Delta ABC#
Draw a line perpendicular from vertex#A# on side#BC# . This is altitude of the triangle#ABC# . This perpendicular also bisects#angle BAC# . As each side of equilateral triangle is#=1# and each angle#=60^@#
Altitude#=1xxcos30^@=sqrt3/2#
Area of#DeltaABC=1/2xx"base"xx"altitude"#
#=1/2xx1xxsqrt3/2=sqrt3/4# 
Area of three squares.
Each square has side#=1# and therefore has area#=1^2=1#
Total area of three squares#=3xx1=3# 
Area of three
#Delta# s#EAF, DBI, HCG#
For#DeltaEAF#
Note that in angle at#A=360^@#
This angle is equal to four angles#=60^@+90^@+90^@+angleEAF#
Equating both we get#angleEAF=360^@240^@=120^@# .
Altitude of#DeltaEAF# can be found as explained in case of#Delta ABC# above
Altitude of#DeltaEAF=1xxcos60^@=1/2#
Half of side#EF=1xxsin60^@=sqrt3/2#
Base#EF=2xxsqrt3/2=sqrt3#
Area of#DeltaEAF=1/2xxsqrt3xx1/2=sqrt3/4#
Similarly area of other two triangles is also same.
Area of hexagon