← Split the surface into the part the unknown touches and the part it does not · Surface Area and Volume of Solids

Split the surface into the part the unknown touches and the part it does not · 12 practice problems

7.G.B.67.G.B.46.EE.B.7

Generated variants — 12

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

Variant 1 easy answer: 7 cm

The cylinder at the right has a surface area of 188.4 cm2188.4\ \text{cm}^2. How tall is it, in cm\text{cm}?

3 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 3 cm has a surface area of 188.4 cm2. We want its height.

Givens
  • The base radius is 3 cm.
  • The surface area is 188.4 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 3 cm, whatever the height.
3×3×3.14×2=56.523 \times 3 \times 3.14 \times 2 = 56.52
56.52 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
188.4−56.52=131.88188.4 - 56.52 = 131.88
131.88 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 18.84 cm.
3×2×3.14=18.843 \times 2 \times 3.14 = 18.84
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
131.88÷18.84=7131.88 \div 18.84 = 7
The cylinder is 7 cm tall.
Answer: 7 cm
4 · Reviewdoes it hold up?

Building it back up: 56.52 for the two bases plus 18.84 times 7 for the side comes to 188.4 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 2 easy answer: 9 cm

The cylinder at the right has a surface area of 326.56 cm2326.56\ \text{cm}^2. How tall is it, in cm\text{cm}?

4 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 4 cm has a surface area of 326.56 cm2. We want its height.

Givens
  • The base radius is 4 cm.
  • The surface area is 326.56 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 4 cm, whatever the height.
4×4×3.14×2=100.484 \times 4 \times 3.14 \times 2 = 100.48
100.48 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
326.56−100.48=226.08326.56 - 100.48 = 226.08
226.08 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 25.12 cm.
4×2×3.14=25.124 \times 2 \times 3.14 = 25.12
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
226.08÷25.12=9226.08 \div 25.12 = 9
The cylinder is 9 cm tall.
Answer: 9 cm
4 · Reviewdoes it hold up?

Building it back up: 100.48 for the two bases plus 25.12 times 9 for the side comes to 326.56 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 3 easy answer: 8 cm

The cylinder at the right has a surface area of 527.52 cm2527.52\ \text{cm}^2. How tall is it, in cm\text{cm}?

6 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 6 cm has a surface area of 527.52 cm2. We want its height.

Givens
  • The base radius is 6 cm.
  • The surface area is 527.52 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 6 cm, whatever the height.
6×6×3.14×2=226.086 \times 6 \times 3.14 \times 2 = 226.08
226.08 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
527.52−226.08=301.44527.52 - 226.08 = 301.44
301.44 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 37.68 cm.
6×2×3.14=37.686 \times 2 \times 3.14 = 37.68
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
301.44÷37.68=8301.44 \div 37.68 = 8
The cylinder is 8 cm tall.
Answer: 8 cm
4 · Reviewdoes it hold up?

Building it back up: 226.08 for the two bases plus 37.68 times 8 for the side comes to 527.52 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 4 easy answer: 12 cm

The cylinder at the right has a surface area of 533.8 cm2533.8\ \text{cm}^2. How tall is it, in cm\text{cm}?

5 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 5 cm has a surface area of 533.8 cm2. We want its height.

Givens
  • The base radius is 5 cm.
  • The surface area is 533.8 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 5 cm, whatever the height.
5×5×3.14×2=1575 \times 5 \times 3.14 \times 2 = 157
157 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
533.8−157=376.8533.8 - 157 = 376.8
376.8 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 31.4 cm.
5×2×3.14=31.45 \times 2 \times 3.14 = 31.4
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
376.8÷31.4=12376.8 \div 31.4 = 12
The cylinder is 12 cm tall.
Answer: 12 cm
4 · Reviewdoes it hold up?

Building it back up: 157 for the two bases plus 31.4 times 12 for the side comes to 533.8 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 5 medium answer: 11 cm

The cylinder at the right has a surface area of 1130.4 cm21130.4\ \text{cm}^2. How tall is it, in cm\text{cm}?

9 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 9 cm has a surface area of 1130.4 cm2. We want its height.

Givens
  • The base radius is 9 cm.
  • The surface area is 1130.4 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 9 cm, whatever the height.
9×9×3.14×2=508.689 \times 9 \times 3.14 \times 2 = 508.68
508.68 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
1130.4−508.68=621.721130.4 - 508.68 = 621.72
621.72 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 56.52 cm.
9×2×3.14=56.529 \times 2 \times 3.14 = 56.52
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
621.72÷56.52=11621.72 \div 56.52 = 11
The cylinder is 11 cm tall.
Answer: 11 cm
4 · Reviewdoes it hold up?

Building it back up: 508.68 for the two bases plus 56.52 times 11 for the side comes to 1130.4 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 6 medium answer: 20 cm

The cylinder at the right has a surface area of 1186.92 cm21186.92\ \text{cm}^2. How tall is it, in cm\text{cm}?

7 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 7 cm has a surface area of 1186.92 cm2. We want its height.

Givens
  • The base radius is 7 cm.
  • The surface area is 1186.92 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 7 cm, whatever the height.
7×7×3.14×2=307.727 \times 7 \times 3.14 \times 2 = 307.72
307.72 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
1186.92−307.72=879.21186.92 - 307.72 = 879.2
879.2 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 43.96 cm.
7×2×3.14=43.967 \times 2 \times 3.14 = 43.96
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
879.2÷43.96=20879.2 \div 43.96 = 20
The cylinder is 20 cm tall.
Answer: 20 cm
4 · Reviewdoes it hold up?

Building it back up: 307.72 for the two bases plus 43.96 times 20 for the side comes to 1186.92 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 7 medium answer: 18 cm

The cylinder at the right has a surface area of 1306.24 cm21306.24\ \text{cm}^2. How tall is it, in cm\text{cm}?

8 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 8 cm has a surface area of 1306.24 cm2. We want its height.

Givens
  • The base radius is 8 cm.
  • The surface area is 1306.24 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 8 cm, whatever the height.
8×8×3.14×2=401.928 \times 8 \times 3.14 \times 2 = 401.92
401.92 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
1306.24−401.92=904.321306.24 - 401.92 = 904.32
904.32 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 50.24 cm.
8×2×3.14=50.248 \times 2 \times 3.14 = 50.24
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
904.32÷50.24=18904.32 \div 50.24 = 18
The cylinder is 18 cm tall.
Answer: 18 cm
4 · Reviewdoes it hold up?

Building it back up: 401.92 for the two bases plus 50.24 times 18 for the side comes to 1306.24 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 8 medium answer: 15 cm

The cylinder at the right has a surface area of 1570 cm21570\ \text{cm}^2. How tall is it, in cm\text{cm}?

10 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 10 cm has a surface area of 1570 cm2. We want its height.

Givens
  • The base radius is 10 cm.
  • The surface area is 1570 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 10 cm, whatever the height.
10×10×3.14×2=62810 \times 10 \times 3.14 \times 2 = 628
628 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
1570−628=9421570 - 628 = 942
942 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 62.8 cm.
10×2×3.14=62.810 \times 2 \times 3.14 = 62.8
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
942÷62.8=15942 \div 62.8 = 15
The cylinder is 15 cm tall.
Answer: 15 cm
4 · Reviewdoes it hold up?

Building it back up: 628 for the two bases plus 62.8 times 15 for the side comes to 1570 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 9 hard answer: 25 cm

The cylinder at the right has a surface area of 2788.32 cm22788.32\ \text{cm}^2. How tall is it, in cm\text{cm}?

12 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 12 cm has a surface area of 2788.32 cm2. We want its height.

Givens
  • The base radius is 12 cm.
  • The surface area is 2788.32 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 12 cm, whatever the height.
12×12×3.14×2=904.3212 \times 12 \times 3.14 \times 2 = 904.32
904.32 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
2788.32−904.32=18842788.32 - 904.32 = 1884
1884 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 75.36 cm.
12×2×3.14=75.3612 \times 2 \times 3.14 = 75.36
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
1884÷75.36=251884 \div 75.36 = 25
The cylinder is 25 cm tall.
Answer: 25 cm
4 · Reviewdoes it hold up?

Building it back up: 904.32 for the two bases plus 75.36 times 25 for the side comes to 2788.32 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 10 hard answer: 30 cm

The cylinder at the right has a surface area of 4239 cm24239\ \text{cm}^2. How tall is it, in cm\text{cm}?

15 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 15 cm has a surface area of 4239 cm2. We want its height.

Givens
  • The base radius is 15 cm.
  • The surface area is 4239 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 15 cm, whatever the height.
15×15×3.14×2=141315 \times 15 \times 3.14 \times 2 = 1413
1413 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
4239−1413=28264239 - 1413 = 2826
2826 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 94.2 cm.
15×2×3.14=94.215 \times 2 \times 3.14 = 94.2
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
2826÷94.2=302826 \div 94.2 = 30
The cylinder is 30 cm tall.
Answer: 30 cm
4 · Reviewdoes it hold up?

Building it back up: 1413 for the two bases plus 94.2 times 30 for the side comes to 4239 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 11 hard answer: 40 cm

The cylinder at the right has a surface area of 7536 cm27536\ \text{cm}^2. How tall is it, in cm\text{cm}?

20 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 20 cm has a surface area of 7536 cm2. We want its height.

Givens
  • The base radius is 20 cm.
  • The surface area is 7536 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 20 cm, whatever the height.
20×20×3.14×2=251220 \times 20 \times 3.14 \times 2 = 2512
2512 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
7536−2512=50247536 - 2512 = 5024
5024 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 125.6 cm.
20×2×3.14=125.620 \times 2 \times 3.14 = 125.6
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
5024÷125.6=405024 \div 125.6 = 40
The cylinder is 40 cm tall.
Answer: 40 cm
4 · Reviewdoes it hold up?

Building it back up: 2512 for the two bases plus 125.6 times 40 for the side comes to 7536 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.
Variant 12 hard answer: 50 cm

The cylinder at the right has a surface area of 11775 cm211775\ \text{cm}^2. How tall is it, in cm\text{cm}?

25 cm ?
Show solution
1 · Understandwhat's really being asked

A cylinder with base radius 25 cm has a surface area of 11775 cm2. We want its height.

Givens
  • The base radius is 25 cm.
  • The surface area is 11775 cm2.
Unknowns
  • The cylinder's height.
Constraints
  • The surface is two bases plus the curved side.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#13 Convert to Algebra

Part of the surface has nothing to do with the height: the two bases depend on the radius alone. Take them off, and what is left is a rectangle whose width is already known.

3 · Execute4 carry out the plan

1Take the two bases off

#16 Count the Complement 7.G.B.4
Each is a circle of radius 25 cm, whatever the height.
25×25×3.14×2=392525 \times 25 \times 3.14 \times 2 = 3925
3925 cm2 that the height cannot change.

2See what is left

#11 Work Backwards 7.G.B.6
Everything else is the curved side.
11775−3925=785011775 - 3925 = 7850
7850 cm2 of curved surface.

3Unroll the side

#13 Convert to Algebra 7.G.B.4
It is a rectangle as wide as the base is round, 157 cm.
25×2×3.14=15725 \times 2 \times 3.14 = 157
Width known, height wanted.

4Divide for the height

#13 Convert to Algebra 6.EE.B.7
Area over width.
7850÷157=507850 \div 157 = 50
The cylinder is 50 cm tall.
Answer: 50 cm
4 · Reviewdoes it hold up?

Building it back up: 3925 for the two bases plus 157 times 50 for the side comes to 11775 cm2, the figure we were given.

Another way: Writing the whole surface as one equation in the height and solving is the same work; taking the bases off first is what stops the height appearing in two places.

Standardsmin grade 7
  • 7.G.B.6 Solve real-world problems involving area, surface area, and volume — Splitting the cylinder's surface into bases and side.
  • 7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base's area and circumference.
  • 6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for the height from the side's area.
💡Takeaway. Find the part of a formula the unknown does not touch and take it away first. What is left is usually one step.