7th Grade Math · Unit 8
When one side of a prism is multiplied by a number, the volume is multiplied by that same number. The surface area changes by a different amount.
Volume scales with that side
If one dimension of a prism is multiplied by a number, the volume is multiplied by that same number.
Doubling the length doubles the volume. Cutting the length in half cuts the volume in half.
Surface area does not copy that factor
The surface area changes, but not by that same factor, because some faces do not use that dimension.
Doubling the length does not double the surface area.
Which faces change
Faces that use the changed side get a new area. Faces that do not use that side stay the same.
Add the new face areas instead of multiplying the old surface area by the factor.
The length is multiplied by 2 or by one half. The volume uses that same factor.
V = lwh
original volume = 6 × 4 × 3 = 72
Multiply the length by 2.
new length = 12
new volume = 12 × 4 × 3 = 144
The volume is multiplied by 2.
72 × 2 = 144
Answer: The volume is multiplied by 2, from 72 to 144 cubic cm.
V = lwh
original volume = 4 × 3 × 2 = 24
Multiply the length by 2.
new length = 8
new volume = 8 × 3 × 2 = 48
The volume is multiplied by 2.
24 × 2 = 48
Answer: The volume is multiplied by 2, from 24 to 48 cubic cm.
V = lwh
original volume = 5 × 2 × 2 = 20
Multiply the length by 2.
new length = 10
new volume = 10 × 2 × 2 = 40
The volume is multiplied by 2.
20 × 2 = 40
Answer: The volume is multiplied by 2, from 20 to 40 cubic cm.
V = lwh
original volume = 8 × 2 × 3 = 48
Multiply the length by 12.
new length = 4
new volume = 4 × 2 × 3 = 24
The volume is multiplied by 12.
48 × 12 = 24
Answer: The volume is multiplied by 12, from 48 to 24 cubic cm.
V = lwh
original volume = 6 × 5 × 2 = 60
Multiply the length by 12.
new length = 3
new volume = 3 × 5 × 2 = 30
The volume is multiplied by 12.
60 × 12 = 30
Answer: The volume is multiplied by 12, from 60 to 30 cubic cm.
V = lwh
original volume = 3 × 4 × 5 = 60
Multiply the length by 2.
new length = 6
new volume = 6 × 4 × 5 = 120
The volume is multiplied by 2.
60 × 2 = 120
Answer: The volume is multiplied by 2, from 60 to 120 cubic cm.
V = lwh
original volume = 10 × 2 × 2 = 40
Multiply the length by 12.
new length = 5
new volume = 5 × 2 × 2 = 20
The volume is multiplied by 12.
40 × 12 = 20
Answer: The volume is multiplied by 12, from 40 to 20 cubic cm.
V = lwh
original volume = 7 × 3 × 2 = 42
Multiply the length by 2.
new length = 14
new volume = 14 × 3 × 2 = 84
The volume is multiplied by 2.
42 × 2 = 84
Answer: The volume is multiplied by 2, from 42 to 84 cubic cm.
V = lwh
original volume = 12 × 2 × 4 = 96
Multiply the length by 12.
new length = 6
new volume = 6 × 2 × 4 = 48
The volume is multiplied by 12.
96 × 12 = 48
Answer: The volume is multiplied by 12, from 96 to 48 cubic cm.
V = lwh
original volume = 9 × 2 × 2 = 36
Multiply the length by 2.
new length = 18
new volume = 18 × 2 × 2 = 72
The volume is multiplied by 2.
36 × 2 = 72
Answer: The volume is multiplied by 2, from 36 to 72 cubic cm.
V = lwh
original volume = 14 × 3 × 2 = 84
Multiply the length by 12.
new length = 7
new volume = 7 × 3 × 2 = 42
The volume is multiplied by 12.
84 × 12 = 42
Answer: The volume is multiplied by 12, from 84 to 42 cubic cm.
The length is multiplied by 3, 4, one third, or one fourth.
V = lwh
original volume = 9 × 2 × 2 = 36
Multiply the length by 13.
new length = 3
new volume = 3 × 2 × 2 = 12
The volume is multiplied by 13.
36 × 13 = 12
Answer: The volume is multiplied by 13, from 36 to 12 cubic cm.
V = lwh
original volume = 2 × 3 × 4 = 24
Multiply the length by 3.
new length = 6
new volume = 6 × 3 × 4 = 72
The volume is multiplied by 3.
24 × 3 = 72
Answer: The volume is multiplied by 3, from 24 to 72 cubic cm.
V = lwh
original volume = 5 × 2 × 2 = 20
Multiply the length by 3.
new length = 15
new volume = 15 × 2 × 2 = 60
The volume is multiplied by 3.
20 × 3 = 60
Answer: The volume is multiplied by 3, from 20 to 60 cubic cm.
V = lwh
original volume = 3 × 4 × 2 = 24
Multiply the length by 4.
new length = 12
new volume = 12 × 4 × 2 = 96
The volume is multiplied by 4.
24 × 4 = 96
Answer: The volume is multiplied by 4, from 24 to 96 cubic cm.
V = lwh
original volume = 1 × 5 × 6 = 30
Multiply the length by 4.
new length = 4
new volume = 4 × 5 × 6 = 120
The volume is multiplied by 4.
30 × 4 = 120
Answer: The volume is multiplied by 4, from 30 to 120 cubic cm.
V = lwh
original volume = 6 × 2 × 5 = 60
Multiply the length by 13.
new length = 2
new volume = 2 × 2 × 5 = 20
The volume is multiplied by 13.
60 × 13 = 20
Answer: The volume is multiplied by 13, from 60 to 20 cubic cm.
V = lwh
original volume = 12 × 3 × 2 = 72
Multiply the length by 13.
new length = 4
new volume = 4 × 3 × 2 = 24
The volume is multiplied by 13.
72 × 13 = 24
Answer: The volume is multiplied by 13, from 72 to 24 cubic cm.
V = lwh
original volume = 8 × 2 × 3 = 48
Multiply the length by 14.
new length = 2
new volume = 2 × 2 × 3 = 12
The volume is multiplied by 14.
48 × 14 = 12
Answer: The volume is multiplied by 14, from 48 to 12 cubic cm.
V = lwh
original volume = 4 × 5 × 2 = 40
Multiply the length by 14.
new length = 1
new volume = 1 × 5 × 2 = 10
The volume is multiplied by 14.
40 × 14 = 10
Answer: The volume is multiplied by 14, from 40 to 10 cubic cm.
V = lwh
original volume = 15 × 2 × 2 = 60
Multiply the length by 13.
new length = 5
new volume = 5 × 2 × 2 = 20
The volume is multiplied by 13.
60 × 13 = 20
Answer: The volume is multiplied by 13, from 60 to 20 cubic cm.
V = lwh
original volume = 16 × 2 × 3 = 96
Multiply the length by 14.
new length = 4
new volume = 4 × 2 × 3 = 24
The volume is multiplied by 14.
96 × 14 = 24
Answer: The volume is multiplied by 14, from 96 to 24 cubic cm.
The surface area changes, but not by the same factor as the length.
SA = 2lw + 2lh + 2wh
original surface area = 158
Multiply the length by 2. The new length is 16 cm.
new surface area = 286
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 286 square cm, compared with 158 square cm
SA = 2lw + 2lh + 2wh
original surface area = 52
Multiply the length by 2. The new length is 8 cm.
new surface area = 92
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 92 square cm, compared with 52 square cm
SA = 2lw + 2lh + 2wh
original surface area = 92
Multiply the length by 12. The new length is 4 cm.
new surface area = 52
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 52 square cm, compared with 92 square cm
SA = 2lw + 2lh + 2wh
original surface area = 94
Multiply the length by 2. The new length is 10 cm.
new surface area = 164
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 164 square cm, compared with 94 square cm
SA = 2lw + 2lh + 2wh
original surface area = 164
Multiply the length by 12. The new length is 5 cm.
new surface area = 94
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 94 square cm, compared with 164 square cm
SA = 2lw + 2lh + 2wh
original surface area = 104
Multiply the length by 2. The new length is 12 cm.
new surface area = 188
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 188 square cm, compared with 104 square cm
SA = 2lw + 2lh + 2wh
original surface area = 188
Multiply the length by 12. The new length is 6 cm.
new surface area = 104
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 104 square cm, compared with 188 square cm
SA = 2lw + 2lh + 2wh
original surface area = 82
Multiply the length by 2. The new length is 14 cm.
new surface area = 152
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 152 square cm, compared with 82 square cm
SA = 2lw + 2lh + 2wh
original surface area = 152
Multiply the length by 12. The new length is 7 cm.
new surface area = 82
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 82 square cm, compared with 152 square cm
SA = 2lw + 2lh + 2wh
original surface area = 124
Multiply the length by 2. The new length is 18 cm.
new surface area = 232
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 232 square cm, compared with 124 square cm
SA = 2lw + 2lh + 2wh
original surface area = 248
Multiply the length by 12. The new length is 8 cm.
new surface area = 136
Only the faces that use the length change, so the surface area is not multiplied by the same factor.
Answer: 136 square cm, compared with 248 square cm
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