What is the surface area, in square centimeters, of a cone with a diameter of 12 feet and a slant height of 14 feet?
We must convert the measurements to the same units. Since 1 foot is equal to 30.48 centimeters, a diameter of 12 feet is 12*30.48=<<12*30.48=365.76>>365.76 centimeters. A slant height of 14 feet is 14*30.48=<<14*30.48=426.72>>426.72 centimeters. The slant height of the cone is, by definition, the hypotenuse of a right triangle whose legs are the radius of the base and the height of the cone. Let $r$ denote the radius of the base. We can see that the height of the cone is the altitude of the right triangle, so by the Pythagorean Theorem, \begin{align*}
r^2+ h^2&=426.72^2\\
r^2+5^2&=426.72^2\\
r^2&=426.72^2-5^2\\
&=181546.4224\\
r&=\sqrt{181546.4224}\\
&\approx426.359,
\end{align*} where $h$ denotes the height of the cone. The formula for the surface area of the cone is $SA_{\text{cone}}=\pi r(r+\ell)$, where $\ell$ denotes the slant height of the cone. Substituting in the given values, we find the surface area of the cone as \begin{align*}
SA_{\text{cone}}&=\pi(426.359)(426.359+426.72)\\
&\approx 287,275.352 \pi.
\end{align*}So, the surface area of the cone is $\boxed{287,275.352 \pi}$ square centimeters.