To find the surface area of a cylinder, you can use the formula:
\[ \text{Surface Area} = 2\pi r^2 + 2\pi rh \]
where:
- \( r \) is the radius of the base,
- \( h \) is the height of the cylinder.
Given a diameter of 8 inches, we can find the radius:
\[ r = \frac{\text{Diameter}}{2} = \frac{8 , \text{in}}{2} = 4 , \text{in} \]
We are also given the height \( h = 10 , \text{in} \).
Now, substitute the values of \( r \) and \( h \) into the surface area formula. First, calculate the area of the two circular bases:
\[ 2\pi r^2 = 2 \times 3.14 \times (4 , \text{in})^2 = 2 \times 3.14 \times 16 , \text{in}^2 = 2 \times 50.24 , \text{in}^2 = 100.48 , \text{in}^2 \]
Next, calculate the lateral surface area:
\[ 2\pi rh = 2 \times 3.14 \times 4 , \text{in} \times 10 , \text{in} = 2 \times 3.14 \times 40 , \text{in}^2 = 2 \times 125.6 , \text{in}^2 = 251.2 , \text{in}^2 \]
Now, add the two areas together for the total surface area:
\[ \text{Total Surface Area} = 100.48 , \text{in}^2 + 251.2 , \text{in}^2 = 351.68 , \text{in}^2 \]
Therefore, the surface area of the cylinder is
\[ \boxed{351.68 , \text{in}^2}. \]