a regular hexagon is inscribed in a circle. The radius of the circle is 18 units. What is the area of the region bounded by the inside of the circle and the outside of the hexagon. Round your answer to the nearest hundredth.

draw the diagonals of the hexagon and you will have 6 equilateral triangles.

let's look at one of these.
the height can found by Pythagoras
h^2 + 9^2 = 18^2
h^2 = √243 = 9√3
so the area of one of them = (1/2)(18)(9√3) = 81√3
there are 6 of them ,so total area of hexagon = 486√3
Area of circle = π(18)^2 = 324π

area of difference = 324π - 81√3 or appr. 176.1

To find the area of the region bounded by the inside of the circle and the outside of the hexagon, we need to calculate the area of the circle and subtract the area of the hexagon.

1. Area of the Circle:
The formula for the area of a circle is A = π * r^2, where A is the area and r is the radius.
Substituting the given radius of 18 units into the formula, we have:
A = π * 18^2

2. Area of the Hexagon:
A regular hexagon can be divided into six congruent equilateral triangles. The formula for the area of an equilateral triangle is A = (sqrt(3)/4) * s^2, where A is the area and s is the length of the side.
The length of the side of the hexagon is equal to the radius of the circle because each side of the hexagon is also a radius.
Substituting the radius of 18 units into the formula, we have:
A = (sqrt(3)/4) * 18^2

3. Calculating the Areas:
Now we can calculate the areas of the circle and the hexagon:
Area of the Circle = π * 18^2
Area of the Hexagon = (sqrt(3)/4) * 18^2

4. Subtracting the Areas:
Finally, subtract the area of the hexagon from the area of the circle to find the area of the region bounded by the inside of the circle and the outside of the hexagon:
Area = Area of Circle - Area of Hexagon

After calculating the values and subtracting the areas, round the final answer to the nearest hundredth.