To find the surface area of a cone, we use the formula:
\[ \text{Surface Area} = \pi r^2 + \pi r l \]
Where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.
In this problem:
- The height of the cone \( h = 16 \) mm,
- The radius of the base \( r = 14 \) mm.
First, we need to find the slant height \( l \) using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Calculating \( l \):
\[ l = \sqrt{(14)^2 + (16)^2} = \sqrt{196 + 256} = \sqrt{452} \]
Calculating \( \sqrt{452} \):
\[ \sqrt{452} \approx 21.26 , \text{mm} \]
Next, we can substitute \( r \) and \( l \) into the surface area formula.
Calculating the base area:
\[ \pi r^2 = \pi (14)^2 = \pi \cdot 196 \approx 3.14 \cdot 196 \approx 384.64 , \text{mm}^2 \]
Calculating the lateral surface area:
\[ \pi r l = \pi \cdot 14 \cdot 21.26 \approx 3.14 \cdot 14 \cdot 21.26 \]
Calculating \( 14 \cdot 21.26 \):
\[ 14 \cdot 21.26 \approx 297.64 \]
Now substituting it into the equation:
\[ \pi r l \approx 3.14 \cdot 297.64 \approx 934.47 , \text{mm}^2 \]
Now, adding both areas together for total surface area:
\[ \text{Surface Area} \approx 384.64 + 934.47 \approx 1319.11 , \text{mm}^2 \]
Hence, the surface area of the cone is approximately:
\[ \boxed{1319.11} , \text{mm}^2 \]