I understand this as what is the area inside the resuted hexagon
that is not covered by the circles.
54sqrt(3) - 27pi
I understand the question as you do - what is the area of that part of the the hexagon that is not covered by circles. And my answer is approximately the same as yours.
1) The area of each of those circles is 9 pi
We have only one circle that is fully inside the hexagon. The other 6 circles are partially in the hexagon. We have to find what part of those 6 circles is in the hexagon.
2)"6 circles exactly surround a circle" - that means that the 6 circles are tangent to the circle in the middle. Thus, the distance from the center of the middle circle to the centers of the other circles is 2r=6. If we connect the center of the middle circle with the centers of the other 6 circles, we'll see that those segments divide the hexagon into 6 pieces. Since the hexagon is right, those segments are equal.
3) 360/6=60 The angle of vertex O of each of the 6 isosceles triangles is 60 degrees. That means that those triangular parts are actually equilateral triangles. Hence, the sides of the hexagon=6
4)60+60=120 degrees =1/3 of 360
The part of each of the 6 circles that is inside the hexagon is 1/3 of its area=1/3*9pi
5)We have 6 of those circles, thus 6*1/3*9pi=18pi
6)The area of the hexagon that is covvered by circles is
18pi(the area of the parts of the six circles)+9pi(the area of the circle in the middle) = 27pi
7)We can find the area of a right hexagon by finding the area of the six triangles. The altitude of an equilateral triangle divides its side to two equal parts. Hence, we cal find the altitude with the Pitagorean theorem. 3^2 + a^2=6^2
The area of each of those 6 triangles is 6*5/2=15
since we have 6 of those, the area of hexagon is 15*6=90
8)The area of the hexagon that is not coverred by circles is