Answer:
[tex]27\sqrt{3}[/tex] square units
Step-by-step explanation:
We need to find the area of the hexagon and then the area of the triangle and subtract the triangle from the hexagon.
The area of a regular hexagon is denoted by the formula: [tex]A=\frac{3\sqrt{3} }{2} s^2[/tex], where s is the side length. Here, s = 6 so:
[tex]A=\frac{3\sqrt{3} }{2} s^2[/tex]
[tex]A=\frac{3\sqrt{3} }{2} *6^2=54\sqrt{3}[/tex]
Now find the area of the triangle. We need just a side length of it. Drop a perpendicular from on of the vertices of the red triangles down to the side of the equilateral triangle. We now have a 30-60-90 triangle with hypotenuse 6. That means that the height (which is half the length of the side of the equilateral triangle) is (6/2) * √3 = 3√3. So, the length of the side is 2 * 3√3 = 6√3.
The area of an equilateral triangle is denoted by: [tex]A=\frac{\sqrt{3} }{4} s^2[/tex]. Here, s = 6√3, so the area is:
[tex]A=\frac{\sqrt{3} }{4} s^2[/tex]
[tex]A=\frac{\sqrt{3} }{4} (6\sqrt{3}) ^2=27\sqrt{3}[/tex]
Now subtract this from the area of the hexagon:
54√3 - 27√3 = 27√3 units squared
