← Each edge appears in two views, so three views give three numbers · Surface Area and Volume of Solids

Each edge appears in two views, so three views give three numbers · 12 practice problems

6.G.A.4

Generated variants — 12

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

Variant 1 easy answer: 214 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 5 by 6, 5 by 7 and 6 by 7. We want its surface area.

Givens

The top view is 5 cm by 6 cm.
The front view is 5 cm by 7 cm.
The side view is 6 cm by 7 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 5 cm edge, the top and side share 6 cm, and the front and side share 7 cm.

5,\ 6,\ 7

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

5 \times 6 = 30,\quad 5 \times 7 = 35,\quad 6 \times 7 = 42

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (30 + 35 + 42) = 214

The surface area is 214 cm2.

Answer: 214 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 30 + 30 + 35 + 35 + 42 + 42, six faces and the same 214 cm2.

Another way: The three edges also give the volume, 5 x 6 x 7 = 210 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 2 easy answer: 132 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 2 by 5, 2 by 8 and 5 by 8. We want its surface area.

Givens

The top view is 2 cm by 5 cm.
The front view is 2 cm by 8 cm.
The side view is 5 cm by 8 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 2 cm edge, the top and side share 5 cm, and the front and side share 8 cm.

2,\ 5,\ 8

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

2 \times 5 = 10,\quad 2 \times 8 = 16,\quad 5 \times 8 = 40

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (10 + 16 + 40) = 132

The surface area is 132 cm2.

Answer: 132 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 10 + 10 + 16 + 16 + 40 + 40, six faces and the same 132 cm2.

Another way: The three edges also give the volume, 2 x 5 x 8 = 80 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 3 easy answer: 228 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 4 by 6, 4 by 9 and 6 by 9. We want its surface area.

Givens

The top view is 4 cm by 6 cm.
The front view is 4 cm by 9 cm.
The side view is 6 cm by 9 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 4 cm edge, the top and side share 6 cm, and the front and side share 9 cm.

4,\ 6,\ 9

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

4 \times 6 = 24,\quad 4 \times 9 = 36,\quad 6 \times 9 = 54

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (24 + 36 + 54) = 228

The surface area is 228 cm2.

Answer: 228 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 24 + 24 + 36 + 36 + 54 + 54, six faces and the same 228 cm2.

Another way: The three edges also give the volume, 4 x 6 x 9 = 216 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 4 easy answer: 268 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 3 by 8, 3 by 10 and 8 by 10. We want its surface area.

Givens

The top view is 3 cm by 8 cm.
The front view is 3 cm by 10 cm.
The side view is 8 cm by 10 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 3 cm edge, the top and side share 8 cm, and the front and side share 10 cm.

3,\ 8,\ 10

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

3 \times 8 = 24,\quad 3 \times 10 = 30,\quad 8 \times 10 = 80

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (24 + 30 + 80) = 268

The surface area is 268 cm2.

Answer: 268 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 24 + 24 + 30 + 30 + 80 + 80, six faces and the same 268 cm2.

Another way: The three edges also give the volume, 3 x 8 x 10 = 240 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 5 medium answer: 298 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 4 by 7, 4 by 11 and 7 by 11. We want its surface area.

Givens

The top view is 4 cm by 7 cm.
The front view is 4 cm by 11 cm.
The side view is 7 cm by 11 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 4 cm edge, the top and side share 7 cm, and the front and side share 11 cm.

4,\ 7,\ 11

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

4 \times 7 = 28,\quad 4 \times 11 = 44,\quad 7 \times 11 = 77

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (28 + 44 + 77) = 298

The surface area is 298 cm2.

Answer: 298 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 28 + 28 + 44 + 44 + 77 + 77, six faces and the same 298 cm2.

Another way: The three edges also give the volume, 4 x 7 x 11 = 308 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 6 medium answer: 510 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 7 by 9, 7 by 12 and 9 by 12. We want its surface area.

Givens

The top view is 7 cm by 9 cm.
The front view is 7 cm by 12 cm.
The side view is 9 cm by 12 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 7 cm edge, the top and side share 9 cm, and the front and side share 12 cm.

7,\ 9,\ 12

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

7 \times 9 = 63,\quad 7 \times 12 = 84,\quad 9 \times 12 = 108

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (63 + 84 + 108) = 510

The surface area is 510 cm2.

Answer: 510 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 63 + 63 + 84 + 84 + 108 + 108, six faces and the same 510 cm2.

Another way: The three edges also give the volume, 7 x 9 x 12 = 756 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 7 medium answer: 498 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 6 by 9, 6 by 13 and 9 by 13. We want its surface area.

Givens

The top view is 6 cm by 9 cm.
The front view is 6 cm by 13 cm.
The side view is 9 cm by 13 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 6 cm edge, the top and side share 9 cm, and the front and side share 13 cm.

6,\ 9,\ 13

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

6 \times 9 = 54,\quad 6 \times 13 = 78,\quad 9 \times 13 = 117

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (54 + 78 + 117) = 498

The surface area is 498 cm2.

Answer: 498 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 54 + 54 + 78 + 78 + 117 + 117, six faces and the same 498 cm2.

Another way: The three edges also give the volume, 6 x 9 x 13 = 702 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 8 medium answer: 608 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 6 by 11, 6 by 14 and 11 by 14. We want its surface area.

Givens

The top view is 6 cm by 11 cm.
The front view is 6 cm by 14 cm.
The side view is 11 cm by 14 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 6 cm edge, the top and side share 11 cm, and the front and side share 14 cm.

6,\ 11,\ 14

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

6 \times 11 = 66,\quad 6 \times 14 = 84,\quad 11 \times 14 = 154

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (66 + 84 + 154) = 608

The surface area is 608 cm2.

Answer: 608 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 66 + 66 + 84 + 84 + 154 + 154, six faces and the same 608 cm2.

Another way: The three edges also give the volume, 6 x 11 x 14 = 924 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 9 hard answer: 792 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 8 by 12, 8 by 15 and 12 by 15. We want its surface area.

Givens

The top view is 8 cm by 12 cm.
The front view is 8 cm by 15 cm.
The side view is 12 cm by 15 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 8 cm edge, the top and side share 12 cm, and the front and side share 15 cm.

8,\ 12,\ 15

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

8 \times 12 = 96,\quad 8 \times 15 = 120,\quad 12 \times 15 = 180

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (96 + 120 + 180) = 792

The surface area is 792 cm2.

Answer: 792 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 96 + 96 + 120 + 120 + 180 + 180, six faces and the same 792 cm2.

Another way: The three edges also give the volume, 8 x 12 x 15 = 1440 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 10 hard answer: 996 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 10 by 13, 10 by 16 and 13 by 16. We want its surface area.

Givens

The top view is 10 cm by 13 cm.
The front view is 10 cm by 16 cm.
The side view is 13 cm by 16 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 10 cm edge, the top and side share 13 cm, and the front and side share 16 cm.

10,\ 13,\ 16

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

10 \times 13 = 130,\quad 10 \times 16 = 160,\quad 13 \times 16 = 208

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (130 + 160 + 208) = 996

The surface area is 996 cm2.

Answer: 996 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 130 + 130 + 160 + 160 + 208 + 208, six faces and the same 996 cm2.

Another way: The three edges also give the volume, 10 x 13 x 16 = 2080 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 11 hard answer: 1172 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 9 by 14, 9 by 20 and 14 by 20. We want its surface area.

Givens

The top view is 9 cm by 14 cm.
The front view is 9 cm by 20 cm.
The side view is 14 cm by 20 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 9 cm edge, the top and side share 14 cm, and the front and side share 20 cm.

9,\ 14,\ 20

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

9 \times 14 = 126,\quad 9 \times 20 = 180,\quad 14 \times 20 = 280

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (126 + 180 + 280) = 1172

The surface area is 1172 cm2.

Answer: 1172 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 126 + 126 + 180 + 180 + 280 + 280, six faces and the same 1172 cm2.

Another way: The three edges also give the volume, 9 x 14 x 20 = 2520 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.

Variant 12 hard answer: 1932 cm²

A rectangular prism looks like this from the top, the front and the side. What is its surface area, in $\text{cm}^2$ ?

Show solution

1 · Understandwhat's really being asked

Three views of a prism measure 12 by 18, 12 by 25 and 18 by 25. We want its surface area.

Givens

The top view is 12 cm by 18 cm.
The front view is 12 cm by 25 cm.
The side view is 18 cm by 25 cm.

Unknowns

The surface area of the prism.

Constraints

The three views are of the same solid, so shared edges match.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

Six numbers are printed but only three are different: every edge shows up in two views. Read off the three edges first, and the six faces become three areas taken twice.

3 · Execute3 carry out the plan

1Match the shared edges

#17 Visualize Spatial Relationships 6.G.A.4

The top and front views share the 12 cm edge, the top and side share 18 cm, and the front and side share 25 cm.

12,\ 18,\ 25

Three edges, not six.

2Work out one of each face

#2 Make a Systematic List 6.G.A.4

Each view is one of the prism's faces.

12 \times 18 = 216,\quad 12 \times 25 = 300,\quad 18 \times 25 = 450

Three different faces.

3Double each and add

#15 Organize Information in More Ways 6.G.A.4

Opposite faces are congruent, so every one of the three appears twice.

2 \times (216 + 300 + 450) = 1932

The surface area is 1932 cm2.

Answer: 1932 cm²

4 · Reviewdoes it hold up?

Counting the faces one at a time gives 216 + 216 + 300 + 300 + 450 + 450, six faces and the same 1932 cm2.

Another way: The three edges also give the volume, 12 x 18 x 25 = 5400 cm3 -- a reminder that the views pin the solid down completely.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Recovering the prism from its views and totalling its faces.

💡Takeaway. Three views of a box repeat each other on purpose. The repeats are how you know which lengths are the same.