Answer:
A = 27(2√3-π) cm² ≈ 8.71 cm²
Step-by-step explanation:
Area of shaded region it is area of hexagon minus area of circle.
A regular hexagon is comprised of six equilateral triangles (of the same sides).
So its area: [tex]A_1=6\cdot\dfrac{S^2\sqrt3}{4}=\dfrac{3S^2\sqrt3}2[/tex] {S = side of the triangle}
Height (H) of such a triangle is equal to radius (R) of a circle inscribed in the hexagon:
[tex]R = H = \dfrac{S\sqrt3}{2}[/tex]
Area of shaded region:
[tex]A=A_1-A_\circ=\dfrac{3S^2\sqrt3}2-\pi R^2=\dfrac{6S^2\sqrt3}4-\pi\left(\dfrac{S\sqrt3}2\right)^2=\dfrac{S^2(6\sqrt3-3\pi)}4[/tex]
S = 6 cm
so:
[tex]A=\dfrac{6^2(6\sqrt3-3\pi)}4=\dfrac{36(6\sqrt3-3\pi)}4=9(6\sqrt3-3\pi)=27(2\sqrt3-\pi)\ cm^2\\\\A=27(2\sqrt3-\pi)\ cm^2\approx8.71\ cm^2[/tex]
