Question

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

Answer 1

Answer:

Area of the shaded region = 3.46 square units

Step-by-step explanation:

Measure of the interior angle of a regular polygon = $\frac{(n - 2)\times180}{n}$

From the picture attached,

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

Interior angle (∠BAF) of the given polygon = $\frac{(6 - 2)\times 180}{6}$

                                                                       = 120°

Measure of ∠BAC = $\frac{120}{4}$

                              = 30°

Now we apply sine rule in ΔAGB,

sin(30°) = $\frac{\text{Opposite side}}{\text{Hypotenuse}}$

            = $\frac{BG}{AB}$

BG = AB[sin(30°)]

     = $2\times \frac{1}{2}$

     = 1

By applying cosine rule in ΔABG,

cos(30°) = $\frac{\text{Adjacent side}}{\text{Hypotenuse}}$

              = $\frac{AG}{AB}$

AG = AB[cos(30°)]

     = $2\times \frac{\sqrt{3} }{2}$

     = $\sqrt{3}$

AC = 2(AG)

     = 2√3

Area of ΔABC = $\frac{1}{2}(\text{Base})(\text{Height})$

                       = $\frac{1}{2}(AC)(BG)$

                       = $\frac{1}{2}(2\sqrt{3} )(1)$

                       = $\sqrt{3}$

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

                                             = 2(Area of ΔABC)

                                             = 2√3

                                             = 3.46 square units