← The height is unchanged, so the volume follows the base · Surface Area and Volume of Solids

The height is unchanged, so the volume follows the base · 12 practice problems

8.G.C.97.G.A.17.G.B.6

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Generated variants — 12

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

Variant 1 easy

A new cylinder is made from the one at the right by quadrupling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 4 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 3 cm and the height 7 cm.
Only the radius changes, multiplied by 4.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 3 cm and 12 cm.

28.26,\quad 452.16

Two circles to compare.

3Compare them

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

Multiplying the radius by 4 multiplies the area by 4 twice over, once for each direction.

452.16 \div 28.26 = 4 \times 4 = 16

16 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

3165.12 \div 197.82 = 16

16 times the volume.

Answer: 16 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 3s and the 7 cancel in the division, so any cylinder whose radius is multiplied by 4 grows by the same 16.

Another way: Answering 4 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 2 easy

A new cylinder is made from the one at the right by quadrupling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 4 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 7 cm and the height 6 cm.
Only the radius changes, multiplied by 4.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 7 cm and 28 cm.

153.86,\quad 2461.76

Two circles to compare.

3Compare them

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

Multiplying the radius by 4 multiplies the area by 4 twice over, once for each direction.

2461.76 \div 153.86 = 4 \times 4 = 16

16 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

14770.56 \div 923.16 = 16

16 times the volume.

Answer: 16 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 7s and the 6 cancel in the division, so any cylinder whose radius is multiplied by 4 grows by the same 16.

Another way: Answering 4 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 3 easy

A new cylinder is made from the one at the right by tripling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 3 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 5 cm and the height 8 cm.
Only the radius changes, multiplied by 3.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 5 cm and 15 cm.

78.5,\quad 706.5

Two circles to compare.

3Compare them

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

Multiplying the radius by 3 multiplies the area by 3 twice over, once for each direction.

706.5 \div 78.5 = 3 \times 3 = 9

9 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

5652 \div 628 = 9

9 times the volume.

Answer: 9 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 5s and the 8 cancel in the division, so any cylinder whose radius is multiplied by 3 grows by the same 9.

Another way: Answering 3 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 4 easy

A new cylinder is made from the one at the right by quadrupling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 4 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 4 cm and the height 9 cm.
Only the radius changes, multiplied by 4.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 4 cm and 16 cm.

50.24,\quad 803.84

Two circles to compare.

3Compare them

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

Multiplying the radius by 4 multiplies the area by 4 twice over, once for each direction.

803.84 \div 50.24 = 4 \times 4 = 16

16 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

7234.56 \div 452.16 = 16

16 times the volume.

Answer: 16 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 4s and the 9 cancel in the division, so any cylinder whose radius is multiplied by 4 grows by the same 16.

Another way: Answering 4 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 5 medium

A new cylinder is made from the one at the right by tripling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 3 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 9 cm and the height 11 cm.
Only the radius changes, multiplied by 3.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 9 cm and 27 cm.

254.34,\quad 2289.06

Two circles to compare.

3Compare them

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

Multiplying the radius by 3 multiplies the area by 3 twice over, once for each direction.

2289.06 \div 254.34 = 3 \times 3 = 9

9 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

25179.66 \div 2797.74 = 9

9 times the volume.

Answer: 9 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 9s and the 11 cancel in the division, so any cylinder whose radius is multiplied by 3 grows by the same 9.

Another way: Answering 3 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 6 medium

A new cylinder is made from the one at the right by doubling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 2 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 6 cm and the height 12 cm.
Only the radius changes, multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 6 cm and 12 cm.

113.04,\quad 452.16

Two circles to compare.

3Compare them

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

Multiplying the radius by 2 multiplies the area by 2 twice over, once for each direction.

452.16 \div 113.04 = 2 \times 2 = 4

4 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

5425.92 \div 1356.48 = 4

4 times the volume.

Answer: 4 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 6s and the 12 cancel in the division, so any cylinder whose radius is multiplied by 2 grows by the same 4.

Another way: Answering 2 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 7 medium

A new cylinder is made from the one at the right by doubling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 2 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 10 cm and the height 13 cm.
Only the radius changes, multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 10 cm and 20 cm.

314,\quad 1256

Two circles to compare.

3Compare them

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

Multiplying the radius by 2 multiplies the area by 2 twice over, once for each direction.

1256 \div 314 = 2 \times 2 = 4

4 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

16328 \div 4082 = 4

4 times the volume.

Answer: 4 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 10s and the 13 cancel in the division, so any cylinder whose radius is multiplied by 2 grows by the same 4.

Another way: Answering 2 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 8 medium

A new cylinder is made from the one at the right by tripling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 3 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 8 cm and the height 15 cm.
Only the radius changes, multiplied by 3.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 8 cm and 24 cm.

200.96,\quad 1808.64

Two circles to compare.

3Compare them

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

Multiplying the radius by 3 multiplies the area by 3 twice over, once for each direction.

1808.64 \div 200.96 = 3 \times 3 = 9

9 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

27129.6 \div 3014.4 = 9

9 times the volume.

Answer: 9 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 8s and the 15 cancel in the division, so any cylinder whose radius is multiplied by 3 grows by the same 9.

Another way: Answering 3 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 9 hard

A new cylinder is made from the one at the right by doubling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 2 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 12 cm and the height 20 cm.
Only the radius changes, multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 12 cm and 24 cm.

452.16,\quad 1808.64

Two circles to compare.

3Compare them

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

Multiplying the radius by 2 multiplies the area by 2 twice over, once for each direction.

1808.64 \div 452.16 = 2 \times 2 = 4

4 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

36172.8 \div 9043.2 = 4

4 times the volume.

Answer: 4 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 12s and the 20 cancel in the division, so any cylinder whose radius is multiplied by 2 grows by the same 4.

Another way: Answering 2 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 10 hard

A new cylinder is made from the one at the right by doubling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 2 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 15 cm and the height 25 cm.
Only the radius changes, multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 15 cm and 30 cm.

706.5,\quad 2826

Two circles to compare.

3Compare them

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

Multiplying the radius by 2 multiplies the area by 2 twice over, once for each direction.

2826 \div 706.5 = 2 \times 2 = 4

4 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

70650 \div 17662.5 = 4

4 times the volume.

Answer: 4 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 15s and the 25 cancel in the division, so any cylinder whose radius is multiplied by 2 grows by the same 4.

Another way: Answering 2 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 11 hard

A new cylinder is made from the one at the right by doubling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 2 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 20 cm and the height 30 cm.
Only the radius changes, multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 20 cm and 40 cm.

1256,\quad 5024

Two circles to compare.

3Compare them

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

Multiplying the radius by 2 multiplies the area by 2 twice over, once for each direction.

5024 \div 1256 = 2 \times 2 = 4

4 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

150720 \div 37680 = 4

4 times the volume.

Answer: 4 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 20s and the 30 cancel in the division, so any cylinder whose radius is multiplied by 2 grows by the same 4.

Another way: Answering 2 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Variant 12 hard

A new cylinder is made from the one at the right by doubling only the radius of its base. The new cylinder's volume is how many times the original cylinder's volume?

Your answer

Show solution

1 · Understandwhat's really being asked

A cylinder's base radius is multiplied by 2 and its height left alone. We want how many times bigger the volume becomes.

Givens

The original radius is 25 cm and the height 40 cm.
Only the radius changes, multiplied by 2.

Unknowns

How many times the volume grows.

Constraints

The height is unchanged.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #17 Visualize Spatial Relationships#8 Analyze the Units

Volume is base area times height, and the height is fixed -- so the whole change is in the base. A circle grows in two directions when its radius does, which is where the squaring comes from.

3 · Execute4 carry out the plan

1See what can change

#8 Analyze the Units 7.G.B.6

Volume is base area times height, and the height stays the same.

V = \text{base} \times \text{height}

Only the base moves.

2Find both bases

#17 Visualize Spatial Relationships 8.G.C.9

Radii 25 cm and 50 cm.

1962.5,\quad 7850

Two circles to compare.

3Compare them

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

Multiplying the radius by 2 multiplies the area by 2 twice over, once for each direction.

7850 \div 1962.5 = 2 \times 2 = 4

4 times the base.

4Carry that to the volume

#8 Analyze the Units 7.G.B.6

The same height on a base that many times larger.

314000 \div 78500 = 4

4 times the volume.

Answer: 4 times

4 · Reviewdoes it hold up?

The starting numbers never mattered: the 25s and the 40 cancel in the division, so any cylinder whose radius is multiplied by 2 grows by the same 4.

Another way: Answering 2 instead treats the cylinder as if it only got wider one way; it grows in two, which is why the factor is squared and the height is not involved at all.

Standardsmin grade 8

8.G.C.9 Know the formulas for volumes of cones, cylinders, and spheres — Using the cylinder's volume rule.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating a scale factor of lengths to one of areas.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Reading the volume as base area times height.

💡Takeaway. Twice the radius is four times the base -- and if the height does not change, four times the volume.

Where next?

← Back to the problem: The height is unchanged, so the volume follows the base