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


See the picture

Find the area of each shaded region each outer polygon is regular See the picture-example-1

1 Answer

8 votes

Answer:

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

Find the area of each shaded region each outer polygon is regular See the picture-example-1
User Binderbound
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