To find the surface area of a cylinder with a base diameter of 6 inches and a height of 6 inches, we need to follow several steps. Let's break down the problem step-by-step:
### Step 1: Determine the Radius
The radius of the base of the cylinder is half of the diameter.
- Given the diameter is 6 inches, the radius [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{6}{2} = 3 \text{ inches} \][/tex]
### Step 2: Calculate the Lateral Surface Area
The formula for the lateral surface area [tex]\( A_{\text{lateral}} \)[/tex] of a cylinder is:
[tex]\[ A_{\text{lateral}} = 2 \pi r h \][/tex]
- Here, [tex]\( r = 3 \)[/tex] inches and [tex]\( h = 6 \)[/tex] inches.
- Plugging in the values:
[tex]\[ A_{\text{lateral}} = 2 \pi (3) (6) = 36 \pi \text{ square inches} \][/tex]
### Step 3: Calculate the Area of the Two Bases
The formula for the area of one base [tex]\( A_{\text{base}} \)[/tex] of a cylinder is:
[tex]\[ A_{\text{base}} = \pi r^2 \][/tex]
- Here [tex]\( r = 3 \)[/tex] inches.
- Plugging in the values:
[tex]\[ A_{\text{base}} = \pi (3)^2 = 9 \pi \text{ square inches} \][/tex]
Since the cylinder has two bases, the total area for the two bases is:
[tex]\[ 2 \times A_{\text{base}} = 2 \times 9 \pi = 18 \pi \text{ square inches} \][/tex]
### Step 4: Calculate the Total Surface Area
The total surface area [tex]\( A_{\text{total}} \)[/tex] of the cylinder is the sum of the lateral surface area and the area of the two bases.
[tex]\[ A_{\text{total}} = A_{\text{lateral}} + 2 \times A_{\text{base}} \][/tex]
- Plugging in the values:
[tex]\[ A_{\text{total}} = 36 \pi + 18 \pi = 54 \pi \text{ square inches} \][/tex]
### Final Answer
The surface area of the cylinder, in terms of [tex]\(\pi\)[/tex], is:
[tex]\[ \boxed{54 \pi \text{ square inches}} \][/tex]
This concludes our step-by-step solution to find the surface area of a cylinder with the given dimensions.
