To calculate the surface area of a cone, we need to know both the radius of the base and the slant height. The formula for the surface area \( A \) of a cone is:
\[ A = \pi r (r + l) \]
where:
- \( r \) is the radius of the base,
- \( l \) is the slant height of the cone.
From the information provided:
- The diameter of the base of the cone is given as 25 cm. The radius \( r \) can be calculated as:
\[ r = \frac{\text{diameter}}{2} = \frac{25 \text{ cm}}{2} = 12.5 \text{ cm} \]
- The height of the cone \( h \) is given as 22 cm. We need to find the slant height \( l \) using the Pythagorean theorem, which is applicable since the height, radius, and slant height form a right triangle:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the values we have:
\[ l = \sqrt{(12.5 \text{ cm})^2 + (22 \text{ cm})^2} \]
Calculating the squares:
\[ (12.5 \text{ cm})^2 = 156.25 \text{ cm}^2 \]
\[ (22 \text{ cm})^2 = 484 \text{ cm}^2 \]
Now add those together:
\[ l = \sqrt{156.25 \text{ cm}^2 + 484 \text{ cm}^2} = \sqrt{640.25 \text{ cm}^2} \approx 25.25 \text{ cm} \]
Now we can calculate the surface area \( A \):
\[ A = \pi r (r + l) \]
Substituting \( r = 12.5 \text{ cm} \) and \( l \approx 25.25 \text{ cm} \):
\[ A = 3.14 \times 12.5 \times (12.5 + 25.25) \]
Calculating \( 12.5 + 25.25 \):
\[ 12.5 + 25.25 = 37.75 \text{ cm} \]
Now substitute that back into the area formula:
\[ A = 3.14 \times 12.5 \times 37.75 \]
Calculating:
\[ 3.14 \times 12.5 \approx 39.25 \]
Now multiply by \( 37.75 \):
\[ A \approx 39.25 \times 37.75 \approx 1484.6875 \text{ cm}^2 \]
Rounded to two decimal places, the surface area of the cone is approximately:
\[ \boxed{1484.69 \text{ cm}^2} \]