Question
EXPLAIN how to find surface area of a rectangular prism. Be specific and use examples.
Answers
GPT-4o mini
To find the surface area of a rectangular prism, you need to understand that a rectangular prism has six rectangular faces. The surface area is the total area of all these faces.
### Steps to Calculate Surface Area of a Rectangular Prism:
1. **Identify Dimensions**:
- Let the length be \( l \).
- Let the width be \( w \).
- Let the height be \( h \).
2. **Calculate Area of Each Face**:
- There are three unique pairs of faces in a rectangular prism:
- Two faces with dimensions \( l \) and \( w \) (the bottom and top).
- Two faces with dimensions \( l \) and \( h \) (the front and back).
- Two faces with dimensions \( w \) and \( h \) (the left and right sides).
3. **Calculate the Areas**:
- Area of the two \( l \times w \) faces:
\[
2 \times (l \times w)
\]
- Area of the two \( l \times h \) faces:
\[
2 \times (l \times h)
\]
- Area of the two \( w \times h \) faces:
\[
2 \times (w \times h)
\]
4. **Add All Areas Together**:
- The total surface area \( SA \) is the sum of the areas of all faces:
\[
SA = 2lw + 2lh + 2wh
\]
5. **Simplify**:
- This formula can be factored as:
\[
SA = 2(lw + lh + wh)
\]
### Example Calculation:
Suppose you have a rectangular prism with the following dimensions:
- Length \( l = 4 \) units
- Width \( w = 3 \) units
- Height \( h = 5 \) units
1. **Calculate each pair of areas**:
- Area of the two \( l \times w \) faces:
\[
2(lw) = 2(4 \times 3) = 2 \times 12 = 24 \text{ square units}
\]
- Area of the two \( l \times h \) faces:
\[
2(lh) = 2(4 \times 5) = 2 \times 20 = 40 \text{ square units}
\]
- Area of the two \( w \times h \) faces:
\[
2(wh) = 2(3 \times 5) = 2 \times 15 = 30 \text{ square units}
\]
2. **Add all areas**:
\[
SA = 24 + 40 + 30 = 94 \text{ square units}
\]
Thus, the surface area of the rectangular prism is **94 square units**.
### Summary:
To find the surface area of a rectangular prism, determine the dimensions, use the formula \( SA = 2(lw + lh + wh) \), calculate the areas of the respective faces, and sum them up. Each step involves basic multiplication and addition.
### Steps to Calculate Surface Area of a Rectangular Prism:
1. **Identify Dimensions**:
- Let the length be \( l \).
- Let the width be \( w \).
- Let the height be \( h \).
2. **Calculate Area of Each Face**:
- There are three unique pairs of faces in a rectangular prism:
- Two faces with dimensions \( l \) and \( w \) (the bottom and top).
- Two faces with dimensions \( l \) and \( h \) (the front and back).
- Two faces with dimensions \( w \) and \( h \) (the left and right sides).
3. **Calculate the Areas**:
- Area of the two \( l \times w \) faces:
\[
2 \times (l \times w)
\]
- Area of the two \( l \times h \) faces:
\[
2 \times (l \times h)
\]
- Area of the two \( w \times h \) faces:
\[
2 \times (w \times h)
\]
4. **Add All Areas Together**:
- The total surface area \( SA \) is the sum of the areas of all faces:
\[
SA = 2lw + 2lh + 2wh
\]
5. **Simplify**:
- This formula can be factored as:
\[
SA = 2(lw + lh + wh)
\]
### Example Calculation:
Suppose you have a rectangular prism with the following dimensions:
- Length \( l = 4 \) units
- Width \( w = 3 \) units
- Height \( h = 5 \) units
1. **Calculate each pair of areas**:
- Area of the two \( l \times w \) faces:
\[
2(lw) = 2(4 \times 3) = 2 \times 12 = 24 \text{ square units}
\]
- Area of the two \( l \times h \) faces:
\[
2(lh) = 2(4 \times 5) = 2 \times 20 = 40 \text{ square units}
\]
- Area of the two \( w \times h \) faces:
\[
2(wh) = 2(3 \times 5) = 2 \times 15 = 30 \text{ square units}
\]
2. **Add all areas**:
\[
SA = 24 + 40 + 30 = 94 \text{ square units}
\]
Thus, the surface area of the rectangular prism is **94 square units**.
### Summary:
To find the surface area of a rectangular prism, determine the dimensions, use the formula \( SA = 2(lw + lh + wh) \), calculate the areas of the respective faces, and sum them up. Each step involves basic multiplication and addition.