Question

what is the surface area, the front face, the back, top, bottom, left, and right for a rectangular prism 8 feet tall, 3 feet wide, 12 ft long

Answer 1

To calculate the surface area and the areas of each face of a rectangular prism, we need to understand the dimensions you provided:

- Height (h) = 8 feet
- Width (w) = 3 feet
- Length (l) = 12 feet

### Areas of Each Face
1. **Front face** (Width x Height):
\( \text{Area}_{\text{front}} = w \times h = 3 \, \text{ft} \times 8 \, \text{ft} = 24 \, \text{ft}^2 \)

2. **Back face** (Width x Height):
\( \text{Area}_{\text{back}} = w \times h = 3 \, \text{ft} \times 8 \, \text{ft} = 24 \, \text{ft}^2 \)

3. **Top face** (Length x Width):
\( \text{Area}_{\text{top}} = l \times w = 12 \, \text{ft} \times 3 \, \text{ft} = 36 \, \text{ft}^2 \)

4. **Bottom face** (Length x Width):
\( \text{Area}_{\text{bottom}} = l \times w = 12 \, \text{ft} \times 3 \, \text{ft} = 36 \, \text{ft}^2 \)

5. **Left face** (Length x Height):
\( \text{Area}_{\text{left}} = l \times h = 12 \, \text{ft} \times 8 \, \text{ft} = 96 \, \text{ft}^2 \)

6. **Right face** (Length x Height):
\( \text{Area}_{\text{right}} = l \times h = 12 \, \text{ft} \times 8 \, \text{ft} = 96 \, \text{ft}^2 \)

### Total Surface Area
To find the total surface area (SA) of the rectangular prism, we sum the areas of all the faces:

\[
\text{Surface Area} = 2 \times (\text{Area}_{\text{front}} + \text{Area}_{\text{back}} + \text{Area}_{\text{top}} + \text{Area}_{\text{bottom}} + \text{Area}_{\text{left}} + \text{Area}_{\text{right}})
\]

Calculating the total:

\[
\text{Surface Area} = 2 \times (24 + 24 + 36 + 36 + 96 + 96)
\]
\[
= 2 \times (312) = 624 \, \text{ft}^2
\]

### Summary
- Area of Front Face: \(24 \, \text{ft}^2\)
- Area of Back Face: \(24 \, \text{ft}^2\)
- Area of Top Face: \(36 \, \text{ft}^2\)
- Area of Bottom Face: \(36 \, \text{ft}^2\)
- Area of Left Face: \(96 \, \text{ft}^2\)
- Area of Right Face: \(96 \, \text{ft}^2\)
- **Total Surface Area**: \(624 \, \text{ft}^2\)