The surface area of a sphere with radius $r$ is $4\pi r^2$. Including the area of its circular base, what is the total surface area of a hemisphere with radius 6 cm? Express your answer in terms of $\pi$.

[asy]

import markers;
size(150);
import geometry;
draw((0,-7)--(0,-1),Arrow);
draw((10,10)--(5,5),Arrow);
label("half of sphere",(10,10),N);
label("circular base",(0,-7),S);

draw(scale(1,.2)*arc((0,0),10,0,180),dashed);
draw(scale(1,.2)*arc((0,0),10,180,360));
draw(Arc((0,0),10,0,180));

[/asy]
The base of the hemisphere is a circle with radius 6 and area $6^2\pi=36\pi$.  The curved top of the hemisphere has half the surface area of a full sphere, which has surface area $4\pi(6^2)=144\pi$, so the curved top of the hemisphere has $144\pi/2=72\pi$.  The total surface area of the hemisphere is $36\pi+72\pi=\boxed{108\pi}$.
108\pi
Level 3
Geometry
Problem: The surface area of a sphere with radius $r$ is $4\pi r^2$. Including the area of its circular base, what is the total surface area of a hemisphere with radius 6 cm? Express your answer in terms of $\pi$.

[asy]

import markers;
size(150);
import geometry;
draw((0,-7)--(0,-1),Arrow);
draw((10,10)--(5,5),Arrow);
label("half of sphere",(10,10),N);
label("circular base",(0,-7),S);

draw(scale(1,.2)*arc((0,0),10,0,180),dashed);
draw(scale(1,.2)*arc((0,0),10,180,360));
draw(Arc((0,0),10,0,180));

[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the total surface area of a hemisphere, we need to consider both the curved surface area and the area of the circular base. The formula for the surface area of a full sphere is given as \(4\pi r^2\). For a hemisphere, which is half of a sphere, the curved surface area is half of the total surface area of the sphere. Therefore, the curved surface area of the hemisphere is:

\[
\text{Curved Surface Area} = \frac{1}{2} \times 4\pi r^2 = 2\pi r^2
\]

Given that the radius \(r\) is 6 cm, we substitute \(r\) into the formula:

\[
\text{Curved Surface Area} = 2\pi (6)^2 = 2\pi \times 36 = 72\pi
\]

Next, we need to calculate the area of the circular base of the hemisphere. The area \(A\) of a circle is given by the formula:

\[
A = \pi r^2
\]

Substituting the radius of the base, which is also 6 cm, we find:

\[
\text{Area of Base} = \pi (6)^2 = \pi \times 36 = 36\pi
\]

Now, to find the total surface area of the hemisphere, we add the curved surface area and the area of the base:

\[
\text{Total Surface Area} = \text{Curved Surface Area} + \text{Area of Base} = 72\pi + 36\pi = 108\pi
\]

Thus, the final answer is \(108\pi\).<|END_OF_LATENT|> The final answer is $108\pi$.