Question

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


See the picture

Answer 1

Area of the shaded region = 3.46 square units

Explanation:

Measure of the interior angle of a regular polygon =
`((n - 2)*180)/(n)`

From the picture attached,

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

Interior angle (∠BAF) of the given polygon =
`((6 - 2)* 180)/(6)`

= 120°

Measure of ∠BAC =
`(120)/(4)`

= 30°

Now we apply sine rule in ΔAGB,

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

=
`(BG)/(AB)`

BG = AB[sin(30°)]

=
`2* (1)/(2)`

= 1

By applying cosine rule in ΔABG,

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

=
`(AG)/(AB)`

AG = AB[cos(30°)]

=
`2* (√(3) )/(2)`

=
`√(3)`

AC = 2(AG)

= 2√3

Area of ΔABC =
`(1)/(2)(\text{Base})(\text{Height})`

=
`(1)/(2)(AC)(BG)`

=
`(1)/(2)(2√(3) )(1)`

=
`√(3)`

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

= 2(Area of ΔABC)

= 2√3

= 3.46 square units