Question Correction
A circle is inscribed in a regular hexagon with side length 10 feet. What is the area of the shaded region? Recall that in a 30–60–90 triangle, if the shortest leg measures x units, then the longer leg measures [tex]x\sqrt{3}[/tex] units and the hypotenuse measures 2x units.
Answer:
(A)[tex](150\sqrt{3}-75\pi) $ Square Units[/tex]
Step-by-step explanation:
Area of the Shaded region =Area of Hexagon-Area of the Circle
Area of Hexagon
Length of the shorter Leg = x ft
Side Length of the Hexagon =10 feet
Perimeter of the Hexagon = 10*6 =60 feet
Apothem of the Hexagon (Length of the longer leg)
= Â [tex]x\sqrt{3}[/tex] feet
[tex]=5\sqrt{3}$ feet[/tex]
[tex]\text{Area of a Regular hexagon}=\dfrac{1}{2} \times $Perimeter \times $Apothem[/tex]
[tex]=\dfrac{1}{2} \times 60 \times 5\sqrt{3}\\=150\sqrt{3}$ Square feet[/tex]
Area of Circle
The radius of the Circle = Apothem of the Hexagon [tex]=5\sqrt{3}$ feet[/tex]
Area of the Circle
[tex]=(5\sqrt{3})^2 \times \pi\\ =25 \times 3 \times \pi\\=75\pi $ Square feet[/tex]
Therefore:
Area of the Shaded region [tex]= (150\sqrt{3}-75\pi) $ Square feet[/tex]
