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

1 answer

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: \(624 , \text{ft}^2\)