To find the surface area of a cone, you need to know the radius (r), the height (h), and the slant height (l) of the cone. The formula for the total surface area (A) of a cone is:
\[ A = \pi r (r + l) \]
Given the diameter (d) of the base of the cone is 25 cm, the radius (r) can be computed as follows:
\[ r = \frac{d}{2} = \frac{25}{2} = 12.5 \text{ cm} \]
You also provided the height (h) of the cone, which is 22 cm. To find the slant height (l), we can use the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \] \[ l = \sqrt{(12.5)^2 + (22)^2} \] \[ l = \sqrt{156.25 + 484} \] \[ l = \sqrt{640.25} \approx 25.3 \text{ cm} \quad (\text{rounded to one decimal place}) \]
Now we can substitute the values of \( \pi \), \( r \), and \( l \) into the surface area formula:
\[ A = \pi r (r + l) \] \[ A = 3.14 \times 12.5 \times (12.5 + 25.3) \] \[ A = 3.14 \times 12.5 \times 37.8 \] \[ A \approx 3.14 \times 12.5 \times 37.8 \approx 1481.925 \text{ cm}^2 \quad (\text{rounded to three decimal places}) \]
Therefore, the surface area of the cone is approximately \( 1481.93 \text{ cm}^2 \).