Find the area of each shaded region each outer polygon is regular

See the picture

Area of the shaded region = 3.46 square units

Step-by-step explanation:

Measure of the interior angle of a regular polygon =

From the picture attached,

Number of sides of the given polygon 'n' = 6

Interior angle (∠BAF) of the given polygon =

                                                                       = 120°

Measure of ∠BAC =

                              = 30°

Now we apply sine rule in ΔAGB,

sin(30°) =

            =

BG = AB[sin(30°)]

     =

     = 1

By applying cosine rule in ΔABG,

cos(30°) =

              =

AG = AB[cos(30°)]

     =

     =

AC = 2(AG)

     = 2√3

Area of ΔABC =

                       =

                       =

                       =

Area of the shaded region = Area of ΔABC + Area of ΔFED

                                             = 2(Area of ΔABC)

                                             = 2√3

                                             = 3.46 square units

irrational, cannot