← Scaling every edge scales the volume three times over · Surface Area and Volume of Solids

Scaling every edge scales the volume three times over · 12 practice problems

7.G.A.16.G.A.2

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 125 times

If every edge of the cube at the right is made five times in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all made five times. We want how many times bigger the new volume is than the old.

Givens

The original cube has 2 cm edges.
Every edge is multiplied by 5.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

2 \times 2 \times 2 = 8

8 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 5 times as long.

10 \times 10 \times 10 = 1000

1000 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 5 was applied three times over, once for each direction.

1000 \div 8 = 5 \times 5 \times 5 = 125

125 times as much.

Answer: 125 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 5 grows by the same 125, because the 2s cancel in the division.

Another way: Picturing it as packing: the new cube holds 125 copies of the old one, 5 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 2 easy answer: 27 times

If every edge of the cube at the right is tripled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all tripled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 3 cm edges.
Every edge is multiplied by 3.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

3 \times 3 \times 3 = 27

27 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 3 times as long.

9 \times 9 \times 9 = 729

729 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 3 was applied three times over, once for each direction.

729 \div 27 = 3 \times 3 \times 3 = 27

27 times as much.

Answer: 27 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 3 grows by the same 27, because the 3s cancel in the division.

Another way: Picturing it as packing: the new cube holds 27 copies of the old one, 3 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 3 easy answer: 8 times

If every edge of the cube at the right is doubled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all doubled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 4 cm edges.
Every edge is multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

4 \times 4 \times 4 = 64

64 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 2 times as long.

8 \times 8 \times 8 = 512

512 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 2 was applied three times over, once for each direction.

512 \div 64 = 2 \times 2 \times 2 = 8

8 times as much.

Answer: 8 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 2 grows by the same 8, because the 4s cancel in the division.

Another way: Picturing it as packing: the new cube holds 8 copies of the old one, 2 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 4 easy answer: 8 times

If every edge of the cube at the right is doubled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all doubled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 5 cm edges.
Every edge is multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

5 \times 5 \times 5 = 125

125 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 2 times as long.

10 \times 10 \times 10 = 1000

1000 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 2 was applied three times over, once for each direction.

1000 \div 125 = 2 \times 2 \times 2 = 8

8 times as much.

Answer: 8 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 2 grows by the same 8, because the 5s cancel in the division.

Another way: Picturing it as packing: the new cube holds 8 copies of the old one, 2 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 5 medium answer: 64 times

If every edge of the cube at the right is quadrupled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all quadrupled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 6 cm edges.
Every edge is multiplied by 4.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

6 \times 6 \times 6 = 216

216 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 4 times as long.

24 \times 24 \times 24 = 13824

13824 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 4 was applied three times over, once for each direction.

13824 \div 216 = 4 \times 4 \times 4 = 64

64 times as much.

Answer: 64 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 4 grows by the same 64, because the 6s cancel in the division.

Another way: Picturing it as packing: the new cube holds 64 copies of the old one, 4 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 6 medium answer: 27 times

If every edge of the cube at the right is tripled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all tripled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 7 cm edges.
Every edge is multiplied by 3.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

7 \times 7 \times 7 = 343

343 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 3 times as long.

21 \times 21 \times 21 = 9261

9261 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 3 was applied three times over, once for each direction.

9261 \div 343 = 3 \times 3 \times 3 = 27

27 times as much.

Answer: 27 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 3 grows by the same 27, because the 7s cancel in the division.

Another way: Picturing it as packing: the new cube holds 27 copies of the old one, 3 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 7 medium answer: 8 times

If every edge of the cube at the right is doubled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all doubled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 8 cm edges.
Every edge is multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

8 \times 8 \times 8 = 512

512 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 2 times as long.

16 \times 16 \times 16 = 4096

4096 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 2 was applied three times over, once for each direction.

4096 \div 512 = 2 \times 2 \times 2 = 8

8 times as much.

Answer: 8 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 2 grows by the same 8, because the 8s cancel in the division.

Another way: Picturing it as packing: the new cube holds 8 copies of the old one, 2 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 8 medium answer: 64 times

If every edge of the cube at the right is quadrupled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all quadrupled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 9 cm edges.
Every edge is multiplied by 4.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

9 \times 9 \times 9 = 729

729 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 4 times as long.

36 \times 36 \times 36 = 46656

46656 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 4 was applied three times over, once for each direction.

46656 \div 729 = 4 \times 4 \times 4 = 64

64 times as much.

Answer: 64 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 4 grows by the same 64, because the 9s cancel in the division.

Another way: Picturing it as packing: the new cube holds 64 copies of the old one, 4 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 9 hard answer: 27 times

If every edge of the cube at the right is tripled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all tripled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 10 cm edges.
Every edge is multiplied by 3.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

10 \times 10 \times 10 = 1000

1000 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 3 times as long.

30 \times 30 \times 30 = 27000

27000 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 3 was applied three times over, once for each direction.

27000 \div 1000 = 3 \times 3 \times 3 = 27

27 times as much.

Answer: 27 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 3 grows by the same 27, because the 10s cancel in the division.

Another way: Picturing it as packing: the new cube holds 27 copies of the old one, 3 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 10 hard answer: 8 times

If every edge of the cube at the right is doubled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all doubled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 12 cm edges.
Every edge is multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

12 \times 12 \times 12 = 1728

1728 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 2 times as long.

24 \times 24 \times 24 = 13824

13824 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 2 was applied three times over, once for each direction.

13824 \div 1728 = 2 \times 2 \times 2 = 8

8 times as much.

Answer: 8 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 2 grows by the same 8, because the 12s cancel in the division.

Another way: Picturing it as packing: the new cube holds 8 copies of the old one, 2 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 11 hard answer: 125 times

If every edge of the cube at the right is made five times in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all made five times. We want how many times bigger the new volume is than the old.

Givens

The original cube has 15 cm edges.
Every edge is multiplied by 5.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

15 \times 15 \times 15 = 3375

3375 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 5 times as long.

75 \times 75 \times 75 = 421875

421875 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 5 was applied three times over, once for each direction.

421875 \div 3375 = 5 \times 5 \times 5 = 125

125 times as much.

Answer: 125 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 5 grows by the same 125, because the 15s cancel in the division.

Another way: Picturing it as packing: the new cube holds 125 copies of the old one, 5 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.

Variant 12 hard answer: 64 times

If every edge of the cube at the right is quadrupled in length, the new cube's volume is how many times the original cube's volume?

Show solution

1 · Understandwhat's really being asked

A cube's edges are all quadrupled. We want how many times bigger the new volume is than the old.

Givens

The original cube has 20 cm edges.
Every edge is multiplied by 4.

Unknowns

How many times the volume grows.

Constraints

All three edges change by the same factor, so the shape stays a cube.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#5 Look for a Pattern

Volume is three lengths multiplied together, so ask what happens to a product when every factor grows. The starting edge is there to check the answer with, not to reach it.

3 · Execute3 carry out the plan

1Write the old volume

#17 Visualize Spatial Relationships 6.G.A.2

A cube's volume is its edge multiplied by itself three times.

20 \times 20 \times 20 = 8000

8000 cm3 to start with.

2Scale every edge

#9 Solve an Easier Related Problem 7.G.A.1

Each of the three edges becomes 4 times as long.

80 \times 80 \times 80 = 512000

512000 cm3 now.

3Compare the two

#5 Look for a Pattern 7.G.A.1

The factor 4 was applied three times over, once for each direction.

512000 \div 8000 = 4 \times 4 \times 4 = 64

64 times as much.

Answer: 64 times

4 · Reviewdoes it hold up?

The starting edge never mattered: any cube scaled by 4 grows by the same 64, because the 20s cancel in the division.

Another way: Picturing it as packing: the new cube holds 64 copies of the old one, 4 along each direction.

Standardsmin grade 7

7.G.A.1 Solve problems involving scale drawings of geometric figures — Reading what scaling every length does to the solid.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Computing a cube's volume from its edge.

💡Takeaway. Length grows once, area twice, volume three times. A shape that is twice as long is eight times as big.