To find the surface area of a cone, you can 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.
Given Data:
- Diameter of the base of the cone: 25 cm
- Height of the cone: 22 cm
Step 1: Find the radius
The radius \( r \) is half of the diameter:
\[ r = \frac{25}{2} = 12.5 \text{ cm} \]
Step 2: Calculate the slant height \( l \)
To find the slant height, we use the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Where \( h \) is the height of the cone. Plugging in the values:
\[ l = \sqrt{(12.5)^2 + (22)^2} \]
Calculating \( (12.5)^2 \) and \( (22)^2 \):
\[ (12.5)^2 = 156.25 \] \[ (22)^2 = 484 \]
Now, summing those values:
\[ l = \sqrt{156.25 + 484} = \sqrt{640.25} \approx 25.25 \text{ cm} \]
Step 3: Calculate the surface area
Now substitute \( r \) and \( l \) into the surface area formula:
\[ \text{Surface Area} = \pi r^2 + \pi r l \]
Calculating each part separately:
- Calculate \( \pi r^2 \):
\[ \pi r^2 = 3.14 \times (12.5)^2 = 3.14 \times 156.25 \approx 490.625 \text{ cm}^2 \]
- Calculate \( \pi r l \):
\[ \pi r l = 3.14 \times 12.5 \times 25.25 \]
Calculating \( 12.5 \times 25.25 \):
\[ 12.5 \times 25.25 \approx 315.625 \]
Now, substituting this back into the formula:
\[ \pi r l \approx 3.14 \times 315.625 \approx 991.625 \text{ cm}^2 \]
Final Calculation:
Now sum both components:
\[ \text{Surface Area} \approx 490.625 + 991.625 \approx 1482.25 \text{ cm}^2 \]
Thus, the surface area of the cone is approximately:
\[ \boxed{1482.25} \text{ cm}^2 \]