We are asked to calculate the area of a regular hexagon that is constructed around six outer circles, each with radius $r$, that form a flower-like pattern. The key is understanding the relationship between the geometry of the circles and the hexagon.

Key Details:

1. Six outer circles each have a radius $r$.
2. These circles are arranged around a central circle of the same radius $r$, and each outer circle touches the central circle and its two immediate neighbors.
3. A regular hexagon is drawn around the six outer circles, such that its sides are tangent to all the outer circles.

Step-by-Step Solution:

Step 1: Understanding the Geometry

The centers of the six outer circles form a regular hexagon around the central circle. These centers are all equidistant from the center of the central circle and from each other. This is because the six outer circles are tangent to each other and to the central circle.

Step 2: Determining the Distance Between Adjacent Centers

Since each outer circle touches the central circle, the distance between the center of the central circle and the center of any outer circle is $2r$ (because the radii of the central and outer circles are both $r$).

Now, the distance between the centers of any two adjacent outer circles is equal to twice the radius $r$ (since they touch each other), i.e., the distance between adjacent outer circle centers is $2r$.

Thus, the centers of the outer circles form a regular hexagon with side length $2r$.

Step 3: Radius of the Hexagon

The radius of the hexagon is the distance from its center to any of its vertices. Since the vertices of the hexagon coincide with the centers of the outer circles, the radius of the hexagon is equal to the distance from the center of the central circle to the center of any outer circle, which we already determined is $2r$.

Thus, the radius of the hexagon is $2r$.

Step 4: Area of the Hexagon

The formula for the area $A$ of a regular hexagon with a radius $R$ (the distance from its center to a vertex) is given by:

$A = \frac{3\sqrt{3}}{2} R^2$

Substituting $R = 2r$ into this formula:

$A = \frac{3\sqrt{3}}{2} (2r)^2 = \frac{3\sqrt{3}}{2} \times 4r^2 = 6\sqrt{3}r^2$

Final Answer:

The area of the hexagon is $6\sqrt{3}r^2$.

---

Would you like further details on any step or clarification? Here are some related questions:

1. How does the arrangement of the circles affect the geometry of the hexagon?
2. What happens if the radius of the outer circles is different from the radius of the central circle?
3. How would the area of the hexagon change if we used more than six outer circles?
4. How do tangency points impact the relationship between the outer circles and the hexagon?
5. How can we derive the formula for the area of a regular hexagon from its side length?

Tip: Remember, understanding the geometric relationships (like distances between circle centers and tangency points) is crucial when dealing with problems that involve inscribed or circumscribed shapes.