Answer:
Area of the shaded region = 3.46 square units
Step-by-step explanation:
Measure of the interior angle of a regular polygon = [tex]\frac{(n - 2)\times180}{n}[/tex]
From the picture attached,
Number of sides of the given polygon 'n' = 6
Interior angle (∠BAF) of the given polygon = [tex]\frac{(6 - 2)\times 180}{6}[/tex]
= 120°
Measure of ∠BAC = [tex]\frac{120}{4}[/tex]
= 30°
Now we apply sine rule in ΔAGB,
sin(30°) = [tex]\frac{\text{Opposite side}}{\text{Hypotenuse}}[/tex]
= [tex]\frac{BG}{AB}[/tex]
BG = AB[sin(30°)]
= [tex]2\times \frac{1}{2}[/tex]
= 1
By applying cosine rule in ΔABG,
cos(30°) = [tex]\frac{\text{Adjacent side}}{\text{Hypotenuse}}[/tex]
= [tex]\frac{AG}{AB}[/tex]
AG = AB[cos(30°)]
= [tex]2\times \frac{\sqrt{3} }{2}[/tex]
= [tex]\sqrt{3}[/tex]
AC = 2(AG)
= 2√3
Area of ΔABC = [tex]\frac{1}{2}(\text{Base})(\text{Height})[/tex]
= [tex]\frac{1}{2}(AC)(BG)[/tex]
= [tex]\frac{1}{2}(2\sqrt{3} )(1)[/tex]
= [tex]\sqrt{3}[/tex]
Area of the shaded region = Area of ΔABC + Area of ΔFED
= 2(Area of ΔABC)
= 2√3
= 3.46 square units
