← A cylinder needs two numbers, and its views give both · Net and Solid Structure

A cylinder needs two numbers, and its views give both · 12 practice problems

7.G.A.37.G.B.47.G.B.6

Generated variants — 12

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

Variant 1 easy answer: 138.16 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 4 cm across from above and 4 by 9 cm from the front. We want its surface area.

Givens

The top view is a circle 4 cm across.
The front view is 4 cm wide and 9 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 4 cm across.

4 \div 2 = 2

The radius is 2 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 9

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

2 \times 2 \times 3.14 \times 2 = 25.12

25.12 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

12.56 \times 9 = 113.04, \quad 25.12 + 113.04 = 138.16

138.16 cm2 in all.

Answer: 138.16 cm²

4 · Reviewdoes it hold up?

The curved side, 113.04 cm2, is larger than the two bases together, 25.12 cm2 -- which fits a cylinder 9 cm tall on a 4 cm base.

Another way: The front view's area, 36 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 2 easy answer: 471 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 10 cm across from above and 10 by 10 cm from the front. We want its surface area.

Givens

The top view is a circle 10 cm across.
The front view is 10 cm wide and 10 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 10 cm across.

10 \div 2 = 5

The radius is 5 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 10

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

5 \times 5 \times 3.14 \times 2 = 157

157 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

31.4 \times 10 = 314, \quad 157 + 314 = 471

471 cm2 in all.

Answer: 471 cm²

4 · Reviewdoes it hold up?

The curved side, 314 cm2, is larger than the two bases together, 157 cm2 -- which fits a cylinder 10 cm tall on a 10 cm base.

Another way: The front view's area, 100 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 3 easy answer: 401.92 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 8 cm across from above and 8 by 12 cm from the front. We want its surface area.

Givens

The top view is a circle 8 cm across.
The front view is 8 cm wide and 12 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 8 cm across.

8 \div 2 = 4

The radius is 4 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 12

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

4 \times 4 \times 3.14 \times 2 = 100.48

100.48 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

25.12 \times 12 = 301.44, \quad 100.48 + 301.44 = 401.92

401.92 cm2 in all.

Answer: 401.92 cm²

4 · Reviewdoes it hold up?

The curved side, 301.44 cm2, is larger than the two bases together, 100.48 cm2 -- which fits a cylinder 12 cm tall on a 8 cm base.

Another way: The front view's area, 96 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 4 easy answer: 527.52 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 12 cm across from above and 12 by 8 cm from the front. We want its surface area.

Givens

The top view is a circle 12 cm across.
The front view is 12 cm wide and 8 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 12 cm across.

12 \div 2 = 6

The radius is 6 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 8

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

6 \times 6 \times 3.14 \times 2 = 226.08

226.08 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

37.68 \times 8 = 301.44, \quad 226.08 + 301.44 = 527.52

527.52 cm2 in all.

Answer: 527.52 cm²

4 · Reviewdoes it hold up?

The curved side, 301.44 cm2, is larger than the two bases together, 226.08 cm2 -- which fits a cylinder 8 cm tall on a 12 cm base.

Another way: The front view's area, 96 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 5 medium answer: 339.12 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 6 cm across from above and 6 by 15 cm from the front. We want its surface area.

Givens

The top view is a circle 6 cm across.
The front view is 6 cm wide and 15 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 6 cm across.

6 \div 2 = 3

The radius is 3 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 15

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

3 \times 3 \times 3.14 \times 2 = 56.52

56.52 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

18.84 \times 15 = 282.6, \quad 56.52 + 282.6 = 339.12

339.12 cm2 in all.

Answer: 339.12 cm²

4 · Reviewdoes it hold up?

The curved side, 282.6 cm2, is larger than the two bases together, 56.52 cm2 -- which fits a cylinder 15 cm tall on a 6 cm base.

Another way: The front view's area, 90 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 6 medium answer: 1299.96 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 18 cm across from above and 18 by 14 cm from the front. We want its surface area.

Givens

The top view is a circle 18 cm across.
The front view is 18 cm wide and 14 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 18 cm across.

18 \div 2 = 9

The radius is 9 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 14

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

9 \times 9 \times 3.14 \times 2 = 508.68

508.68 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

56.52 \times 14 = 791.28, \quad 508.68 + 791.28 = 1299.96

1299.96 cm2 in all.

Answer: 1299.96 cm²

4 · Reviewdoes it hold up?

The curved side, 791.28 cm2, is larger than the two bases together, 508.68 cm2 -- which fits a cylinder 14 cm tall on a 18 cm base.

Another way: The front view's area, 252 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 7 medium answer: 1186.92 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 14 cm across from above and 14 by 20 cm from the front. We want its surface area.

Givens

The top view is a circle 14 cm across.
The front view is 14 cm wide and 20 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 14 cm across.

14 \div 2 = 7

The radius is 7 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 20

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

7 \times 7 \times 3.14 \times 2 = 307.72

307.72 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

43.96 \times 20 = 879.2, \quad 307.72 + 879.2 = 1186.92

1186.92 cm2 in all.

Answer: 1186.92 cm²

4 · Reviewdoes it hold up?

The curved side, 879.2 cm2, is larger than the two bases together, 307.72 cm2 -- which fits a cylinder 20 cm tall on a 14 cm base.

Another way: The front view's area, 280 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 8 medium answer: 1004.8 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 20 cm across from above and 20 by 6 cm from the front. We want its surface area.

Givens

The top view is a circle 20 cm across.
The front view is 20 cm wide and 6 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 20 cm across.

20 \div 2 = 10

The radius is 10 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 6

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

10 \times 10 \times 3.14 \times 2 = 628

628 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

62.8 \times 6 = 376.8, \quad 628 + 376.8 = 1004.8

1004.8 cm2 in all.

Answer: 1004.8 cm²

4 · Reviewdoes it hold up?

The curved side, 376.8 cm2, is smaller than the two bases together, 628 cm2 -- which fits a cylinder 6 cm tall on a 20 cm base.

Another way: The front view's area, 120 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 9 hard answer: 1657.92 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 16 cm across from above and 16 by 25 cm from the front. We want its surface area.

Givens

The top view is a circle 16 cm across.
The front view is 16 cm wide and 25 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 16 cm across.

16 \div 2 = 8

The radius is 8 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 25

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

8 \times 8 \times 3.14 \times 2 = 401.92

401.92 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

50.24 \times 25 = 1256, \quad 401.92 + 1256 = 1657.92

1657.92 cm2 in all.

Answer: 1657.92 cm²

4 · Reviewdoes it hold up?

The curved side, 1256 cm2, is larger than the two bases together, 401.92 cm2 -- which fits a cylinder 25 cm tall on a 16 cm base.

Another way: The front view's area, 400 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 10 hard answer: 3165.12 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 24 cm across from above and 24 by 30 cm from the front. We want its surface area.

Givens

The top view is a circle 24 cm across.
The front view is 24 cm wide and 30 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 24 cm across.

24 \div 2 = 12

The radius is 12 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 30

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

12 \times 12 \times 3.14 \times 2 = 904.32

904.32 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

75.36 \times 30 = 2260.8, \quad 904.32 + 2260.8 = 3165.12

3165.12 cm2 in all.

Answer: 3165.12 cm²

4 · Reviewdoes it hold up?

The curved side, 2260.8 cm2, is larger than the two bases together, 904.32 cm2 -- which fits a cylinder 30 cm tall on a 24 cm base.

Another way: The front view's area, 720 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 11 hard answer: 3108.6 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 30 cm across from above and 30 by 18 cm from the front. We want its surface area.

Givens

The top view is a circle 30 cm across.
The front view is 30 cm wide and 18 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 30 cm across.

30 \div 2 = 15

The radius is 15 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 18

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

15 \times 15 \times 3.14 \times 2 = 1413

1413 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

94.2 \times 18 = 1695.6, \quad 1413 + 1695.6 = 3108.6

3108.6 cm2 in all.

Answer: 3108.6 cm²

4 · Reviewdoes it hold up?

The curved side, 1695.6 cm2, is larger than the two bases together, 1413 cm2 -- which fits a cylinder 18 cm tall on a 30 cm base.

Another way: The front view's area, 540 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.

Variant 12 hard answer: 5275.2 cm²

The figures at the right show a cylinder seen from above and from the front. What is the cylinder's surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

A cylinder is 40 cm across from above and 40 by 22 cm from the front. We want its surface area.

Givens

The top view is a circle 40 cm across.
The front view is 40 cm wide and 22 cm tall.

Unknowns

The cylinder's surface area.

Constraints

Both views are of the same cylinder, so their widths agree.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#15 Organize Information in More Ways

A cylinder is fixed by a radius and a height, and no more. Read those two off the views -- the circle gives one, the rectangle's height the other -- and the surface follows from the usual two parts.

3 · Execute4 carry out the plan

1Read the radius from above

#17 Visualize Spatial Relationships 7.G.A.3

The top view is the base itself, 40 cm across.

40 \div 2 = 20

The radius is 20 cm.

2Read the height from the front

#15 Organize Information in More Ways 7.G.A.3

The front view is the cylinder squashed flat: its width is the diameter again, and its height is the cylinder's.

\text{height} = 22

The two views agree on the width.

3Add the two bases

#7 Identify Subproblems 7.G.B.4

Top and bottom, each a circle of that radius.

20 \times 20 \times 3.14 \times 2 = 2512

2512 cm2 of flat surface.

4Unroll the side and add it

#7 Identify Subproblems 7.G.B.6

The curved side is a rectangle as wide as the base is round.

125.6 \times 22 = 2763.2, \quad 2512 + 2763.2 = 5275.2

5275.2 cm2 in all.

Answer: 5275.2 cm²

4 · Reviewdoes it hold up?

The curved side, 2763.2 cm2, is larger than the two bases together, 2512 cm2 -- which fits a cylinder 22 cm tall on a 40 cm base.

Another way: The front view's area, 880 cm2, is the cylinder's flat cross-section, not part of its surface -- a useful reminder that a view is not always an area you can use.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the solid's measurements off its views.
7.G.B.4 Know the formulas for area and circumference of a circle — Finding the base circle's area and circumference.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Totalling the cylinder's surface.

💡Takeaway. Two views of a cylinder repeat its width on purpose. That repeat is how you know they describe the same solid.