Find volume and surface area of each figure.
Worksheet with 10 numbered diagrams of rectangular prisms and cubes, each labeled with dimensions in centimeters, asking to find volume and surface area.
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Step-by-step solution for: Volume and Surface Area of Rectangular Prisms Two worksheets 1 ...
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Show Answer Key & Explanations
Step-by-step solution for: Volume and Surface Area of Rectangular Prisms Two worksheets 1 ...
To solve the problem of finding the volume and surface area of each rectangular prism shown in the image, we will use the following formulas:
1. Volume of a Rectangular Prism:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
2. Surface Area of a Rectangular Prism:
\[
SA = 2(\text{length} \times \text{width} + \text{width} \times \text{height} + \text{height} \times \text{length})
\]
#### Figure 1:
- Dimensions: \(7 \, \text{cm} \times 40 \, \text{cm} \times 30 \, \text{cm}\)
- Volume:
\[
V = 7 \times 40 \times 30 = 8400 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(7 \times 40 + 40 \times 30 + 30 \times 7) = 2(280 + 1200 + 210) = 2 \times 1690 = 3380 \, \text{cm}^2
\]
#### Figure 2:
- Dimensions: \(47 \, \text{cm} \times 45 \, \text{cm} \times 45 \, \text{cm}\)
- Volume:
\[
V = 47 \times 45 \times 45 = 93187.5 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(47 \times 45 + 45 \times 45 + 45 \times 47) = 2(2115 + 2025 + 2115) = 2 \times 6255 = 12510 \, \text{cm}^2
\]
#### Figure 3:
- Dimensions: \(20 \, \text{cm} \times 20 \, \text{cm} \times 20 \, \text{cm}\)
- Volume:
\[
V = 20 \times 20 \times 20 = 8000 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(20 \times 20 + 20 \times 20 + 20 \times 20) = 2(400 + 400 + 400) = 2 \times 1200 = 2400 \, \text{cm}^2
\]
#### Figure 4:
- Dimensions: \(90 \, \text{cm} \times 43 \, \text{cm} \times 24 \, \text{cm}\)
- Volume:
\[
V = 90 \times 43 \times 24 = 95040 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(90 \times 43 + 43 \times 24 + 24 \times 90) = 2(3870 + 1032 + 2160) = 2 \times 7062 = 14124 \, \text{cm}^2
\]
#### Figure 5:
- Dimensions: \(38 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm}\)
- Volume:
\[
V = 38 \times 3 \times 3 = 342 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(38 \times 3 + 3 \times 3 + 3 \times 38) = 2(114 + 9 + 114) = 2 \times 237 = 474 \, \text{cm}^2
\]
#### Figure 6:
- Dimensions: \(25 \, \text{cm} \times 25 \, \text{cm} \times 25 \, \text{cm}\)
- Volume:
\[
V = 25 \times 25 \times 25 = 15625 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(25 \times 25 + 25 \times 25 + 25 \times 25) = 2(625 + 625 + 625) = 2 \times 1875 = 3750 \, \text{cm}^2
\]
#### Figure 7:
- Dimensions: \(33 \, \text{cm} \times 33 \, \text{cm} \times 33 \, \text{cm}\)
- Volume:
\[
V = 33 \times 33 \times 33 = 35937 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(33 \times 33 + 33 \times 33 + 33 \times 33) = 2(1089 + 1089 + 1089) = 2 \times 3267 = 6534 \, \text{cm}^2
\]
#### Figure 8:
- Dimensions: \(40 \, \text{cm} \times 8 \, \text{cm} \times 8 \, \text{cm}\)
- Volume:
\[
V = 40 \times 8 \times 8 = 2560 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(40 \times 8 + 8 \times 8 + 8 \times 40) = 2(320 + 64 + 320) = 2 \times 704 = 1408 \, \text{cm}^2
\]
#### Figure 9:
- Dimensions: \(2 \, \text{cm} \times 49 \, \text{cm} \times 49 \, \text{cm}\)
- Volume:
\[
V = 2 \times 49 \times 49 = 4802 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(2 \times 49 + 49 \times 49 + 49 \times 2) = 2(98 + 2401 + 98) = 2 \times 2597 = 5194 \, \text{cm}^2
\]
#### Figure 10:
- Dimensions: \(22 \, \text{cm} \times 22 \, \text{cm} \times 22 \, \text{cm}\)
- Volume:
\[
V = 22 \times 22 \times 22 = 10648 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(22 \times 22 + 22 \times 22 + 22 \times 22) = 2(484 + 484 + 484) = 2 \times 1452 = 2904 \, \text{cm}^2
\]
\[
\boxed{
\begin{array}{ll}
\text{Figure 1:} & V = 8400 \, \text{cm}^3, \, SA = 3380 \, \text{cm}^2 \\
\text{Figure 2:} & V = 93187.5 \, \text{cm}^3, \, SA = 12510 \, \text{cm}^2 \\
\text{Figure 3:} & V = 8000 \, \text{cm}^3, \, SA = 2400 \, \text{cm}^2 \\
\text{Figure 4:} & V = 95040 \, \text{cm}^3, \, SA = 14124 \, \text{cm}^2 \\
\text{Figure 5:} & V = 342 \, \text{cm}^3, \, SA = 474 \, \text{cm}^2 \\
\text{Figure 6:} & V = 15625 \, \text{cm}^3, \, SA = 3750 \, \text{cm}^2 \\
\text{Figure 7:} & V = 35937 \, \text{cm}^3, \, SA = 6534 \, \text{cm}^2 \\
\text{Figure 8:} & V = 2560 \, \text{cm}^3, \, SA = 1408 \, \text{cm}^2 \\
\text{Figure 9:} & V = 4802 \, \text{cm}^3, \, SA = 5194 \, \text{cm}^2 \\
\text{Figure 10:} & V = 10648 \, \text{cm}^3, \, SA = 2904 \, \text{cm}^2 \\
\end{array}
}
\]
Formulas:
1. Volume of a Rectangular Prism:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
2. Surface Area of a Rectangular Prism:
\[
SA = 2(\text{length} \times \text{width} + \text{width} \times \text{height} + \text{height} \times \text{length})
\]
Step-by-Step Solution for Each Figure:
#### Figure 1:
- Dimensions: \(7 \, \text{cm} \times 40 \, \text{cm} \times 30 \, \text{cm}\)
- Volume:
\[
V = 7 \times 40 \times 30 = 8400 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(7 \times 40 + 40 \times 30 + 30 \times 7) = 2(280 + 1200 + 210) = 2 \times 1690 = 3380 \, \text{cm}^2
\]
#### Figure 2:
- Dimensions: \(47 \, \text{cm} \times 45 \, \text{cm} \times 45 \, \text{cm}\)
- Volume:
\[
V = 47 \times 45 \times 45 = 93187.5 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(47 \times 45 + 45 \times 45 + 45 \times 47) = 2(2115 + 2025 + 2115) = 2 \times 6255 = 12510 \, \text{cm}^2
\]
#### Figure 3:
- Dimensions: \(20 \, \text{cm} \times 20 \, \text{cm} \times 20 \, \text{cm}\)
- Volume:
\[
V = 20 \times 20 \times 20 = 8000 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(20 \times 20 + 20 \times 20 + 20 \times 20) = 2(400 + 400 + 400) = 2 \times 1200 = 2400 \, \text{cm}^2
\]
#### Figure 4:
- Dimensions: \(90 \, \text{cm} \times 43 \, \text{cm} \times 24 \, \text{cm}\)
- Volume:
\[
V = 90 \times 43 \times 24 = 95040 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(90 \times 43 + 43 \times 24 + 24 \times 90) = 2(3870 + 1032 + 2160) = 2 \times 7062 = 14124 \, \text{cm}^2
\]
#### Figure 5:
- Dimensions: \(38 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm}\)
- Volume:
\[
V = 38 \times 3 \times 3 = 342 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(38 \times 3 + 3 \times 3 + 3 \times 38) = 2(114 + 9 + 114) = 2 \times 237 = 474 \, \text{cm}^2
\]
#### Figure 6:
- Dimensions: \(25 \, \text{cm} \times 25 \, \text{cm} \times 25 \, \text{cm}\)
- Volume:
\[
V = 25 \times 25 \times 25 = 15625 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(25 \times 25 + 25 \times 25 + 25 \times 25) = 2(625 + 625 + 625) = 2 \times 1875 = 3750 \, \text{cm}^2
\]
#### Figure 7:
- Dimensions: \(33 \, \text{cm} \times 33 \, \text{cm} \times 33 \, \text{cm}\)
- Volume:
\[
V = 33 \times 33 \times 33 = 35937 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(33 \times 33 + 33 \times 33 + 33 \times 33) = 2(1089 + 1089 + 1089) = 2 \times 3267 = 6534 \, \text{cm}^2
\]
#### Figure 8:
- Dimensions: \(40 \, \text{cm} \times 8 \, \text{cm} \times 8 \, \text{cm}\)
- Volume:
\[
V = 40 \times 8 \times 8 = 2560 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(40 \times 8 + 8 \times 8 + 8 \times 40) = 2(320 + 64 + 320) = 2 \times 704 = 1408 \, \text{cm}^2
\]
#### Figure 9:
- Dimensions: \(2 \, \text{cm} \times 49 \, \text{cm} \times 49 \, \text{cm}\)
- Volume:
\[
V = 2 \times 49 \times 49 = 4802 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(2 \times 49 + 49 \times 49 + 49 \times 2) = 2(98 + 2401 + 98) = 2 \times 2597 = 5194 \, \text{cm}^2
\]
#### Figure 10:
- Dimensions: \(22 \, \text{cm} \times 22 \, \text{cm} \times 22 \, \text{cm}\)
- Volume:
\[
V = 22 \times 22 \times 22 = 10648 \, \text{cm}^3
\]
- Surface Area:
\[
SA = 2(22 \times 22 + 22 \times 22 + 22 \times 22) = 2(484 + 484 + 484) = 2 \times 1452 = 2904 \, \text{cm}^2
\]
Final Answers:
\[
\boxed{
\begin{array}{ll}
\text{Figure 1:} & V = 8400 \, \text{cm}^3, \, SA = 3380 \, \text{cm}^2 \\
\text{Figure 2:} & V = 93187.5 \, \text{cm}^3, \, SA = 12510 \, \text{cm}^2 \\
\text{Figure 3:} & V = 8000 \, \text{cm}^3, \, SA = 2400 \, \text{cm}^2 \\
\text{Figure 4:} & V = 95040 \, \text{cm}^3, \, SA = 14124 \, \text{cm}^2 \\
\text{Figure 5:} & V = 342 \, \text{cm}^3, \, SA = 474 \, \text{cm}^2 \\
\text{Figure 6:} & V = 15625 \, \text{cm}^3, \, SA = 3750 \, \text{cm}^2 \\
\text{Figure 7:} & V = 35937 \, \text{cm}^3, \, SA = 6534 \, \text{cm}^2 \\
\text{Figure 8:} & V = 2560 \, \text{cm}^3, \, SA = 1408 \, \text{cm}^2 \\
\text{Figure 9:} & V = 4802 \, \text{cm}^3, \, SA = 5194 \, \text{cm}^2 \\
\text{Figure 10:} & V = 10648 \, \text{cm}^3, \, SA = 2904 \, \text{cm}^2 \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of volume and surface area of rectangular prism worksheet.