In the diagram, the smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is $6.$ What is the area of the shaded region?

[asy]
size(100);
import graph;
filldraw(Circle((0,0),2),mediumgray);
filldraw(Circle((-1,0),1),white);
filldraw(Circle((1,0),1),white);
[/asy]
Label the center of the larger circle $O$ and the points of contact between the larger circle and the smaller circles $A$ and $B.$ Draw the radius $OA$ of the larger circle.

[asy]
size(120);
import graph;
filldraw(Circle((0,0),2),mediumgray);
filldraw(Circle((-1,0),1),white);
filldraw(Circle((1,0),1),white);
draw((-2,0)--(0,0));

label("$A$",(-2,0),W); label("$O$",(0,0),E); label("$B$",(2,0),E);

[/asy]

Since the smaller circle and the larger circle touch at $A,$ the diameter through $A$ of the smaller circle lies along the diameter through $A$ of the larger circle. (This is because each diameter is perpendicular to the common tangent at the point of contact.)

Since $AO$ is a radius of the larger circle, it is a diameter of the smaller circle.

Since the radius of the larger circle is $6,$ the diameter of the smaller circle is $6,$ so the radius of the smaller circle on the left is $3.$

Similarly, we can draw a radius through $O$ and $B$ and deduce that the radius of the smaller circle on the right is also $3.$ The area of the shaded region equals the area of the larger circle minus the combined area of the two smaller circles. Thus, the area of the shaded region is $$6^2\pi - 3^2\pi - 3^2\pi = 36\pi - 9\pi - 9\pi = \boxed{18\pi}.$$
18\pi
Level 2
Geometry
Problem: In the diagram, the smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is $6.$ What is the area of the shaded region?

[asy]
size(100);
import graph;
filldraw(Circle((0,0),2),mediumgray);
filldraw(Circle((-1,0),1),white);
filldraw(Circle((1,0),1),white);
[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the area of the shaded region, we first need to determine the area of the larger circle. The formula for the area of a circle is A = πr². Given that the radius of the larger circle is 6, we calculate its area as follows:

A_larger = π(6)² = π(36) = 36π.

Next, we need to analyze the smaller circles. The problem states that the smaller circles touch each other at the center of the larger circle. This implies that the centers of the smaller circles are located at the center of the larger circle, and they are tangent to the larger circle.

Since the smaller circles touch the larger circle and each other, we can deduce that the radius of each smaller circle must be half the radius of the larger circle. Therefore, the radius of each smaller circle is:

r_smaller = 6 / 2 = 3.

Now, we calculate the area of one smaller circle using the same area formula:

A_smaller = π(3)² = π(9) = 9π.

Since there are two smaller circles, we find the total area of the smaller circles:

Total area of smaller circles = 2 * A_smaller = 2 * 9π = 18π.

Finally, to find the area of the shaded region, we subtract the total area of the smaller circles from the area of the larger circle:

Area_shaded = A_larger - Total area of smaller circles = 36π - 18π = 18π.

Thus, the final answer is 18π.<|END_OF_LATENT|> The final answer is $18\pi$.