To solve the given problem, let's work through the steps in detail.
### 1. Determine the Side Length of the Square Base
We start with the given information:
- Height of the prism, \( h = 13 \) cm
- Lateral area of the prism, \( A_L = 624 \) cm²
We need to find the side length \( a \) of the square base.
#### Step 1: Using the Lateral Area Formula
For a regular quadrilateral (square) prism, the lateral area \( A_L \) can be expressed as:
[tex]\[ A_L = 4 \cdot a \cdot h \][/tex]
Solving for \( a \):
[tex]\[ a = \frac{A_L}{4 \cdot h} \][/tex]
[tex]\[ a = \frac{624}{4 \cdot 13} \][/tex]
[tex]\[ a = \frac{624}{52} \][/tex]
[tex]\[ a = 12 \text{ cm} \][/tex]
So, the side length \( a \) of the square base is \( 12 \) cm.
### 2. Calculate the Surface Area of the Prism
The surface area \( A \) of the prism includes the area of the two square bases and the lateral area.
#### Step 2: Area of One Square Base
The area of one square base (\( A_{\text{base}} \)) is:
[tex]\[ A_{\text{base}} = a^2 \][/tex]
[tex]\[ A_{\text{base}} = 12^2 \][/tex]
[tex]\[ A_{\text{base}} = 144 \text{ cm}^2 \][/tex]
#### Step 3: Total Surface Area
The total surface area \( A \) of the prism is:
[tex]\[ A = 2 \cdot A_{\text{base}} + A_L \][/tex]
[tex]\[ A = 2 \cdot 144 + 624 \][/tex]
[tex]\[ A = 288 + 624 \][/tex]
[tex]\[ A = 912 \text{ cm}^2 \][/tex]
So, the surface area of the prism is \( 912 \) cm².
### 3. Calculate the Volume of the Prism
The volume (\( V \)) of the prism can be calculated using the area of the base and the height.
#### Step 4: Volume of the Prism
The volume \( V \) is:
[tex]\[ V = A_{\text{base}} \cdot h \][/tex]
[tex]\[ V = 144 \cdot 13 \][/tex]
[tex]\[ V = 1872 \text{ cm}^3 \][/tex]
So, the volume of the prism is \( 1872 \) cm³.
### Summary
1. The surface area of the prism is \( 912 \) cm².
2. The volume of the prism is [tex]\( 1872 \)[/tex] cm³.
