Question

Find the total outer surface area of a cylinder having a radius of 4 cm and height of 6 cm and is open at the top

Answer 1

To find the total outer surface area of a cylinder that is open at the top, we need to consider both the lateral surface area and the area of the base.

1. **Lateral Surface Area**: The formula for the lateral surface area of a cylinder is given by:

\[
\text{Lateral Surface Area} = 2\pi rh
\]

where \( r \) is the radius and \( h \) is the height.

For our cylinder:
- \( r = 4 \, \text{cm} \)
- \( h = 6 \, \text{cm} \)

Substituting these values into the formula:

\[
\text{Lateral Surface Area} = 2\pi (4 \, \text{cm})(6 \, \text{cm}) = 48\pi \, \text{cm}^2
\]

2. **Area of the Base**: The area of the base is given by the formula for the area of a circle:

\[
\text{Base Area} = \pi r^2
\]

Substituting the radius:

\[
\text{Base Area} = \pi (4 \, \text{cm})^2 = 16\pi \, \text{cm}^2
\]

3. **Total Outer Surface Area**: Since the cylinder is open at the top, the total outer surface area is the sum of the lateral surface area and the area of the base:

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

\[
\text{Total Surface Area} = 48\pi \, \text{cm}^2 + 16\pi \, \text{cm}^2 = 64\pi \, \text{cm}^2
\]

Thus, the total outer surface area of the cylinder is

\[
\boxed{64\pi \, \text{cm}^2}
\]

If you want a numerical approximation, substituting \(\pi \approx 3.14\):

\[
64\pi \approx 64 \times 3.14 \approx 200.96 \, \text{cm}^2
\]

So, in approximation, the total outer surface area is about

\[
\boxed{200.96 \, \text{cm}^2}
\]