What is the area, in square units, of a regular hexagon inscribed in a circle whose area is $324\pi$ square units? Express your answer in simplest radical form.
Note that since the area is $\pi r^2 = 324 \pi$, where $r$ is the radius, we must have $r=\sqrt{324}=18$. Thus the distance from the center of the hexagon to a vertex is $18$, and we can break up the hexagon into $6$ equilateral triangles, each of which has side length $18$. The area of an equilateral triangle of side length $s$ is $\frac{s^2 \sqrt{3}}{4}$, so the area of each equilateral triangle is $81 \sqrt{3}$, making the total $6(81 \sqrt{3}) = \boxed{486 \sqrt{3}}$.
486\sqrt{3}
Level 4
Geometry
Problem: What is the area, in square units, of a regular hexagon inscribed in a circle whose area is $324\pi$ square units? Express your answer in simplest radical form.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the area of the regular hexagon, we first need to determine the radius of the circle in which it is inscribed. The area of the circle is given as $324\pi$ square units. The formula for the area of a circle is $A = \pi r^2$. Setting this equal to the given area, we have:

$324\pi = \pi r^2$.

Dividing both sides by $\pi$ gives:

$324 = r^2$.

Taking the square root of both sides, we find:

$r = 18$.

Next, we use the relationship between the radius of the circumscribed circle and the side length of the hexagon. For a regular hexagon, the radius of the circumscribed circle is equal to the length of a side of the hexagon. Therefore, the side length $s$ of the hexagon is also $18$.

The area $A$ of a regular hexagon can be calculated using the formula:

$A = \frac{3\sqrt{3}}{2} s^2$.

Substituting $s = 18$ into the formula gives:

$A = \frac{3\sqrt{3}}{2} (18)^2$.

Calculating $(18)^2$ results in $324$. Thus, we have:

$A = \frac{3\sqrt{3}}{2} \cdot 324$.

Now, simplifying this expression:

$A = \frac{3 \cdot 324 \sqrt{3}}{2} = \frac{972 \sqrt{3}}{2} = 486\sqrt{3}$.

Therefore, the area of the regular hexagon is $486\sqrt{3}$ square units.<|END_OF_LATENT|> The final answer is $486\sqrt{3}$.