A cone is generated by rotating triangle $ABC$ around side $\overline{AB}$. Its total surface area is $\pi$ times what number?
1 answer
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 $8 \pi + \pi r^2 = 8 \pi + \pi (4/\pi)^2 = \boxed{24}$.