To find the surface area of a regular prism where the base is a regular quadrilateral (square) with side length \(4 \, \text{cm}\) and height \(6 \, \text{cm}\), we need to calculate several components:
1. Area of the base
2. Perimeter of the base
3. Lateral surface area
4. Total surface area
### Step 1: Calculate the area of the base
Since the base is a square:
[tex]\[ \text{Area of the base} = \text{side length}^2 = 4 \, \text{cm} \times 4 \, \text{cm} = 16 \, \text{cm}^2 \][/tex]
### Step 2: Calculate the perimeter of the base
For a square, the perimeter is given by:
[tex]\[ \text{Perimeter of the base} = 4 \times \text{side length} = 4 \times 4 \, \text{cm} = 16 \, \text{cm} \][/tex]
### Step 3: Calculate the lateral surface area
The lateral surface area of a prism is found by multiplying the perimeter of the base by the height of the prism:
[tex]\[ \text{Lateral surface area} = \text{Perimeter of the base} \times \text{height} = 16 \, \text{cm} \times 6 \, \text{cm} = 96 \, \text{cm}^2 \][/tex]
### Step 4: Calculate the total surface area
The total surface area of the prism is the sum of the areas of both bases and the lateral surface area. Since the prism has two identical bases:
[tex]\[ \text{Total surface area} = 2 \times \text{Area of the base} + \text{Lateral surface area} \][/tex]
Thus:
[tex]\[ \text{Total surface area} = 2 \times 16 \, \text{cm}^2 + 96 \, \text{cm}^2 = 32 \, \text{cm}^2 + 96 \, \text{cm}^2 = 128 \, \text{cm}^2 \][/tex]
Therefore, the surface area of the prism is [tex]\(128 \, \text{cm}^2\)[/tex].
