A cone is generated by rotating triangle $ABC$ around side $\overline{AB}$. Its total surface area is $\pi$ times what number?
The slant height of the cone equals the distance from $C$ to side $\overline{AB}$, which is 8. The circumference of the base of the cone is 4 (since it circumscribes a circle with diameter 4). Therefore, $\pi r = 4$, so the radius $r=4/\pi$. The lateral surface area is $\pi r \cdot 8 = 8 \pi^2/ \pi = 8 \pi$. This gives a total surface area of $8 \pi + \pi r^2 = 8 \pi + \pi (4/\pi)^2 = \boxed{24}$.
To find the total surface area of the cone, we need to find the area of the base and the lateral surface area, and then add them together.
The base of the cone is the circle created by rotating side $\overline{AB}$. The radius of this circle is the length of the altitude from $C$ to $\overline{AB}$. Let's call this altitude $h$.
Now, let's find the lateral surface area of the cone. This is the surface area of the curved part of the cone, not including the base. The lateral surface area can be found by multiplying the slant height $l$ of the cone by the circumference $c$ of the base (circle).
The slant height $l$ of the cone is the length of the hypotenuse of right triangle $ABC$. Using the Pythagorean theorem, we can find $l$ in terms of $h$ and the length of $\overline{BC}$.
The length of $\overline{BC}$ is the base of the triangle $ABC$, which is the circumference $c$ of the base of the cone.
Now, let's summarize what we know:
- The radius of the base of the cone is $h$
- The slant height of the cone is $l$, which can be found using the Pythagorean theorem
- The circumference of the base of the cone is $c$, which is the length of $\overline{BC}$
The total surface area of the cone is the sum of the area of the base and the lateral surface area:
\[ \text{Total Surface Area} = \text{Area of Base} + \text{Lateral Surface Area} \]
\[ \text{Total Surface Area} = \pi h^2 + \pi c l \]
Since we are given that the total surface area is $\pi$ times some number, we can write:
\[ \pi \cdot \text{Total Surface Area} = \pi ( \pi h^2 + \pi c l) \]
\[ \pi \cdot \text{Total Surface Area} = \pi^2 h^2 + \pi^2 c l \]
Therefore, the total surface area of the cone is $\pi$ times $(\pi h^2 + \pi c l)$.