Find the surface area of a cylinder with a base diameter of 6 ft and a height of 6 ft. Write your answer in terms of π, and be sure to include the correct unit.

To find the surface area of a cylinder, we need to consider both the lateral surface area and the area of the two bases.

Given:
- Diameter of the base (\(d\)) = 6 ft
- Radius of the base (\(r\)) = \(\frac{d}{2} = \frac{6}{2} = 3\) ft
- Height of the cylinder (\(h\)) = 6 ft

### Step 1: Calculate the lateral surface area
The lateral surface area of a cylinder is given by the formula:
\[ \text{Lateral Surface Area} = 2\pi rh \]

Substituting the values:
\[ \text{Lateral Surface Area} = 2\pi (3 \text{ ft})(6 \text{ ft}) = 36\pi \text{ ft}^2 \]

### Step 2: Calculate the area of the two bases
The area of one base is given by the formula:
\[ \text{Area of one base} = \pi r^2 \]

Substituting the radius:
\[ \text{Area of one base} = \pi (3 \text{ ft})^2 = 9\pi \text{ ft}^2 \]

Since there are two bases, the total area of the bases is:
\[ \text{Total Area of the Bases} = 2 \times 9\pi \text{ ft}^2 = 18\pi \text{ ft}^2 \]

### Step 3: Calculate the total surface area
The total surface area of the cylinder is the sum of the lateral surface area and the total area of the bases:
\[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Total Area of the Bases} \]
\[ \text{Total Surface Area} = 36\pi \text{ ft}^2 + 18\pi \text{ ft}^2 = 54\pi \text{ ft}^2 \]

Therefore, the surface area of the cylinder is:
\[ 54\pi \text{ ft}^2 \]

Would you like more details or have any questions about this calculation?

Here are 8 relative questions that expand on this topic:
1. How do you derive the formula for the lateral surface area of a cylinder?
2. If the height of the cylinder were doubled, how would that affect the surface area?
3. How does the surface area of a cylinder change if only the diameter is doubled?
4. What is the volume of this cylinder?
5. How do you find the surface area of a cylinder without using \(\pi\)?
6. Can you explain the difference between lateral surface area and total surface area for a cylinder?
7. How would you find the surface area of a hollow cylinder?
8. How do the formulas for the surface area of a cylinder compare to those of a cone?

**Tip:** Always keep track of your units throughout the calculation to ensure accuracy and correctness in your final answer.

Learn how to calculate the surface area of a cylinder with a base diameter of 6 ft and a height of 6 ft using π. Includes step-by-step solution and formula application for lateral and base surface areas. Suitable for students in Grades 9-12.

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