Certainly! To calculate the total surface area of a solid cylinder, we need to add together the areas of the two circular bases and the area of the rectangular side which wraps around the height of the cylinder. Let's calculate it step by step:
1. Area of one circular base:
The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
where \( r \) is the radius of the circle. For a cylinder with radius \( 3 \) cm, the area of one base is:
\[ \pi \times 3^2 = 9\pi \text{ cm}^2 \]
2. Area of two circular bases:
Since the cylinder has two identical bases, we need to multiply the area of one base by 2 to get the combined area of both bases:
\[ 2 \times 9\pi = 18\pi \text{ cm}^2 \]
3. Area of the side (lateral surface area):
The lateral surface area of a cylinder can be thought of as the area of a rectangle that wraps around the outside of the cylinder. The width of this rectangle is the circumference of the base (given by \( 2\pi r \)) and the length is the height \( h \) of the cylinder. Thus, the formula for the lateral surface area is:
\[ \text{Lateral surface area} = 2\pi r h \]
For the given cylinder, we plug in \( r = 3 \) cm and \( h = 4 \) cm to get:
\[ 2\pi \times 3 \times 4 = 24\pi \text{ cm}^2 \]
4. Total surface area:
Now, to find the total surface area of the cylinder, we add the area of both bases to the lateral surface area:
\[ \text{Total surface area} = \text{Area of two bases} + \text{Lateral surface area} \]
\[ \text{Total surface area} = 18\pi + 24\pi = 42\pi \text{ cm}^2 \]
Thus, expressed in terms of \( \pi \), the total surface area of the cylinder is \( 42\pi \) square centimeters. However, please note that there seems to be a discrepancy cause the surface areas described initially are not in terms of \( \pi \) like the answer requires, but rather in their full decimal forms. If you use these forms, the base area is \( \sim56.55 \), the side area is \( \sim75.40 \), and the total surface area is \( \sim131.95 \) square centimeters.
