Question

A cone shows a base diameter of 25 centimeters, 22 centimeters as the hypotenuse or the side of the cone.What is the surface area of the cone? Use 3.14 for pi

Answer 1

To calculate the surface area of a cone, we need to find both the lateral surface area and the base area. The formula for the surface area of a cone is:

\[
\text{Surface Area} = \text{Lateral Surface Area} + \text{Base Area}
\]

### Step 1: Calculate the radius and height
1. **Radius (r)**: The diameter is given as 25 cm, so the radius is:
\[
r = \frac{\text{diameter}}{2} = \frac{25}{2} = 12.5 \text{ cm}
\]

2. **Hypotenuse (slant height, l)**: The slant height is given as 22 cm.

3. **Height (h)**: We can find the height of the cone using the Pythagorean theorem. The relationship is:
\[
l^2 = r^2 + h^2
\]
Plugging in the values:
\[
22^2 = 12.5^2 + h^2
\]
\[
484 = 156.25 + h^2
\]
\[
h^2 = 484 - 156.25 = 327.75
\]
\[
h = \sqrt{327.75} \approx 18.11 \text{ cm}
\]

### Step 2: Calculate the lateral surface area
The formula for the lateral surface area of a cone is:
\[
\text{Lateral Surface Area} = \pi r l
\]
Substituting the known values:
\[
\text{Lateral Surface Area} = 3.14 \times 12.5 \times 22
\]
Calculating:
\[
\text{Lateral Surface Area} = 3.14 \times 275 = 865.5 \text{ cm}^2
\]

### Step 3: Calculate the base area
The formula for the area of the base (which is a circle) is:
\[
\text{Base Area} = \pi r^2
\]
Substituting the known values:
\[
\text{Base Area} = 3.14 \times (12.5)^2
\]
Calculating:
\[
\text{Base Area} = 3.14 \times 156.25 = 490.625 \text{ cm}^2
\]

### Step 4: Calculate the total surface area
Now, combine the lateral surface area and the base area:
\[
\text{Surface Area} = \text{Lateral Surface Area} + \text{Base Area}
\]
\[
\text{Surface Area} = 865.5 + 490.625 = 1356.125 \text{ cm}^2
\]

### Final Result
The surface area of the cone is approximately:
\[
\boxed{1356.13 \text{ cm}^2}
\]