Problem Description:
The task is to calculate the
volume and
surface area of rectangular prisms (cuboids) for four different shapes. The dimensions of each prism are provided in the image.
Key Formulas:
1.
Volume of a Rectangular Prism:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
2.
Surface Area of a Rectangular Prism:
\[
SA = 2 \times (\text{length} \times \text{width} + \text{width} \times \text{height} + \text{height} \times \text{length})
\]
Step-by-Step Solution:
####
Problem 1:
- Dimensions:
- Length (\( l \)) = 3 in
- Width (\( w \)) = 3 in
- Height (\( h \)) = 2 in
Volume:
\[
V = l \times w \times h = 3 \times 3 \times 2 = 18 \, \text{in}^3
\]
Surface Area:
\[
SA = 2 \times (l \times w + w \times h + h \times l)
\]
\[
SA = 2 \times (3 \times 3 + 3 \times 2 + 2 \times 3)
\]
\[
SA = 2 \times (9 + 6 + 6) = 2 \times 21 = 42 \, \text{in}^2
\]
Answer for Problem 1:
\[
\boxed{18 \, \text{in}^3, 42 \, \text{in}^2}
\]
---
####
Problem 2:
- Dimensions:
- Length (\( l \)) = 4 in
- Width (\( w \)) = 4 in
- Height (\( h \)) = 4 in
This is a cube (a special case of a rectangular prism where all sides are equal).
Volume:
\[
V = l \times w \times h = 4 \times 4 \times 4 = 64 \, \text{in}^3
\]
Surface Area:
For a cube, the surface area formula simplifies to:
\[
SA = 6 \times (\text{side length})^2
\]
\[
SA = 6 \times 4^2 = 6 \times 16 = 96 \, \text{in}^2
\]
Answer for Problem 2:
\[
\boxed{64 \, \text{in}^3, 96 \, \text{in}^2}
\]
---
####
Problem 3:
- Dimensions:
- Length (\( l \)) = 9 in
- Width (\( w \)) = 5 in
- Height (\( h \)) = 6 in
Volume:
\[
V = l \times w \times h = 9 \times 5 \times 6 = 270 \, \text{in}^3
\]
Surface Area:
\[
SA = 2 \times (l \times w + w \times h + h \times l)
\]
\[
SA = 2 \times (9 \times 5 + 5 \times 6 + 6 \times 9)
\]
\[
SA = 2 \times (45 + 30 + 54) = 2 \times 129 = 258 \, \text{in}^2
\]
Answer for Problem 3:
\[
\boxed{270 \, \text{in}^3, 258 \, \text{in}^2}
\]
---
####
Problem 4:
- Dimensions:
- Length (\( l \)) = 7 in
- Width (\( w \)) = 6 in
- Height (\( h \)) = 8 in
Volume:
\[
V = l \times w \times h = 7 \times 6 \times 8 = 336 \, \text{in}^3
\]
Surface Area:
\[
SA = 2 \times (l \times w + w \times h + h \times l)
\]
\[
SA = 2 \times (7 \times 6 + 6 \times 8 + 8 \times 7)
\]
\[
SA = 2 \times (42 + 48 + 56) = 2 \times 146 = 292 \, \text{in}^2
\]
Answer for Problem 4:
\[
\boxed{336 \, \text{in}^3, 292 \, \text{in}^2}
\]
---
Final Answers:
1. \(\boxed{18 \, \text{in}^3, 42 \, \text{in}^2}\)
2. \(\boxed{64 \, \text{in}^3, 96 \, \text{in}^2}\)
3. \(\boxed{270 \, \text{in}^3, 258 \, \text{in}^2}\)
4. \(\boxed{336 \, \text{in}^3, 292 \, \text{in}^2}\)
Parent Tip: Review the logic above to help your child master the concept of additive volume 5th grade worksheet.
