Practice · The sides of a prism are one band as wide as the base's perimeter · Sensim Math

Variant 1 easy

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 14 cm apart, with every corner square. We want its surface area.

Givens

The L is 14 cm along one outer edge and 10 cm along the other.
One arm is 4 cm wide and the other 6 cm.
The two L faces are 14 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 14 by 4, and the standing arm adds 6 by 6 above it.

14 \times 4 + 6 \times 6 = 92

Each L face is 92 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 14 by 10 rectangle, so the distance round is the same as that rectangle's.

(14 + 10) \times 2 = 48

48 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 48 cm wide and 14 cm tall.

48 \times 14 = 672

The sides are 672 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

92 \times 2 + 672 = 856

The surface area is 856 cm2.

Answer: 856 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 48 cm, the same as a 14 by 10 rectangle's.

Another way: Naming all eight side faces separately gives the same 672 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 2 easy

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 12 cm apart, with every corner square. We want its surface area.

Givens

The L is 15 cm along one outer edge and 9 cm along the other.
One arm is 3 cm wide and the other 5 cm.
The two L faces are 12 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 15 by 3, and the standing arm adds 6 by 5 above it.

15 \times 3 + 6 \times 5 = 75

Each L face is 75 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 15 by 9 rectangle, so the distance round is the same as that rectangle's.

(15 + 9) \times 2 = 48

48 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 48 cm wide and 12 cm tall.

48 \times 12 = 576

The sides are 576 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

75 \times 2 + 576 = 726

The surface area is 726 cm2.

Answer: 726 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 48 cm, the same as a 15 by 9 rectangle's.

Another way: Naming all eight side faces separately gives the same 576 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 3 easy

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 18 cm apart, with every corner square. We want its surface area.

Givens

The L is 16 cm along one outer edge and 11 cm along the other.
One arm is 5 cm wide and the other 4 cm.
The two L faces are 18 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 16 by 5, and the standing arm adds 6 by 4 above it.

16 \times 5 + 6 \times 4 = 104

Each L face is 104 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 16 by 11 rectangle, so the distance round is the same as that rectangle's.

(16 + 11) \times 2 = 54

54 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 54 cm wide and 18 cm tall.

54 \times 18 = 972

The sides are 972 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

104 \times 2 + 972 = 1180

The surface area is 1180 cm2.

Answer: 1180 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 54 cm, the same as a 16 by 11 rectangle's.

Another way: Naming all eight side faces separately gives the same 972 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 4 medium

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 15 cm apart, with every corner square. We want its surface area.

Givens

The L is 20 cm along one outer edge and 12 cm along the other.
One arm is 4 cm wide and the other 7 cm.
The two L faces are 15 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 20 by 4, and the standing arm adds 8 by 7 above it.

20 \times 4 + 8 \times 7 = 136

Each L face is 136 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 20 by 12 rectangle, so the distance round is the same as that rectangle's.

(20 + 12) \times 2 = 64

64 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 64 cm wide and 15 cm tall.

64 \times 15 = 960

The sides are 960 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

136 \times 2 + 960 = 1232

The surface area is 1232 cm2.

Answer: 1232 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 64 cm, the same as a 20 by 12 rectangle's.

Another way: Naming all eight side faces separately gives the same 960 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 5 easy

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 20 cm apart, with every corner square. We want its surface area.

Givens

The L is 18 cm along one outer edge and 10 cm along the other.
One arm is 5 cm wide and the other 6 cm.
The two L faces are 20 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 18 by 5, and the standing arm adds 5 by 6 above it.

18 \times 5 + 5 \times 6 = 120

Each L face is 120 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 18 by 10 rectangle, so the distance round is the same as that rectangle's.

(18 + 10) \times 2 = 56

56 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 56 cm wide and 20 cm tall.

56 \times 20 = 1120

The sides are 1120 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

120 \times 2 + 1120 = 1360

The surface area is 1360 cm2.

Answer: 1360 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 56 cm, the same as a 18 by 10 rectangle's.

Another way: Naming all eight side faces separately gives the same 1120 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 6 medium

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 16 cm apart, with every corner square. We want its surface area.

Givens

The L is 22 cm along one outer edge and 15 cm along the other.
One arm is 6 cm wide and the other 9 cm.
The two L faces are 16 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 22 by 6, and the standing arm adds 9 by 9 above it.

22 \times 6 + 9 \times 9 = 213

Each L face is 213 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 22 by 15 rectangle, so the distance round is the same as that rectangle's.

(22 + 15) \times 2 = 74

74 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 74 cm wide and 16 cm tall.

74 \times 16 = 1184

The sides are 1184 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

213 \times 2 + 1184 = 1610

The surface area is 1610 cm2.

Answer: 1610 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 74 cm, the same as a 22 by 15 rectangle's.

Another way: Naming all eight side faces separately gives the same 1184 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 7 medium

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 25 cm apart, with every corner square. We want its surface area.

Givens

The L is 24 cm along one outer edge and 14 cm along the other.
One arm is 6 cm wide and the other 8 cm.
The two L faces are 25 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 24 by 6, and the standing arm adds 8 by 8 above it.

24 \times 6 + 8 \times 8 = 208

Each L face is 208 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 24 by 14 rectangle, so the distance round is the same as that rectangle's.

(24 + 14) \times 2 = 76

76 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 76 cm wide and 25 cm tall.

76 \times 25 = 1900

The sides are 1900 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

208 \times 2 + 1900 = 2316

The surface area is 2316 cm2.

Answer: 2316 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 76 cm, the same as a 24 by 14 rectangle's.

Another way: Naming all eight side faces separately gives the same 1900 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 8 medium

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 22 cm apart, with every corner square. We want its surface area.

Givens

The L is 28 cm along one outer edge and 16 cm along the other.
One arm is 7 cm wide and the other 12 cm.
The two L faces are 22 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 28 by 7, and the standing arm adds 9 by 12 above it.

28 \times 7 + 9 \times 12 = 304

Each L face is 304 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 28 by 16 rectangle, so the distance round is the same as that rectangle's.

(28 + 16) \times 2 = 88

88 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 88 cm wide and 22 cm tall.

88 \times 22 = 1936

The sides are 1936 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

304 \times 2 + 1936 = 2544

The surface area is 2544 cm2.

Answer: 2544 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 88 cm, the same as a 28 by 16 rectangle's.

Another way: Naming all eight side faces separately gives the same 1936 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 9 hard

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 28 cm apart, with every corner square. We want its surface area.

Givens

The L is 26 cm along one outer edge and 20 cm along the other.
One arm is 9 cm wide and the other 11 cm.
The two L faces are 28 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 26 by 9, and the standing arm adds 11 by 11 above it.

26 \times 9 + 11 \times 11 = 355

Each L face is 355 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 26 by 20 rectangle, so the distance round is the same as that rectangle's.

(26 + 20) \times 2 = 92

92 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 92 cm wide and 28 cm tall.

92 \times 28 = 2576

The sides are 2576 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

355 \times 2 + 2576 = 3286

The surface area is 3286 cm2.

Answer: 3286 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 92 cm, the same as a 26 by 20 rectangle's.

Another way: Naming all eight side faces separately gives the same 2576 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 10 hard

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 30 cm apart, with every corner square. We want its surface area.

Givens

The L is 12 cm along one outer edge and 8 cm along the other.
One arm is 3 cm wide and the other 4 cm.
The two L faces are 30 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 12 by 3, and the standing arm adds 5 by 4 above it.

12 \times 3 + 5 \times 4 = 56

Each L face is 56 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 12 by 8 rectangle, so the distance round is the same as that rectangle's.

(12 + 8) \times 2 = 40

40 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 40 cm wide and 30 cm tall.

40 \times 30 = 1200

The sides are 1200 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

56 \times 2 + 1200 = 1312

The surface area is 1312 cm2.

Answer: 1312 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 40 cm, the same as a 12 by 8 rectangle's.

Another way: Naming all eight side faces separately gives the same 1200 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 11 hard

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 10 cm apart, with every corner square. We want its surface area.

Givens

The L is 30 cm along one outer edge and 18 cm along the other.
One arm is 8 cm wide and the other 10 cm.
The two L faces are 10 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 30 by 8, and the standing arm adds 10 by 10 above it.

30 \times 8 + 10 \times 10 = 340

Each L face is 340 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 30 by 18 rectangle, so the distance round is the same as that rectangle's.

(30 + 18) \times 2 = 96

96 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 96 cm wide and 10 cm tall.

96 \times 10 = 960

The sides are 960 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

340 \times 2 + 960 = 1640

The surface area is 1640 cm2.

Answer: 1640 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 96 cm, the same as a 30 by 18 rectangle's.

Another way: Naming all eight side faces separately gives the same 960 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

Variant 12 hard

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A solid has congruent L-shaped top and bottom faces 9 cm apart, with every corner square. We want its surface area.

Givens

The L is 32 cm along one outer edge and 24 cm along the other.
One arm is 10 cm wide and the other 14 cm.
The two L faces are 9 cm apart.

Unknowns

The surface area of the solid.

Constraints

Every pair of meeting edges is perpendicular, so each side face is a rectangle.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

There are eight side faces and naming them one at a time is how one gets lost. The solid is a prism on an L base, so its sides are a single band whose width is the L's perimeter -- trace that once and nothing can be missed.

3 · Execute4 carry out the plan

1Find the L's area

#7 Identify Subproblems 6.G.A.4

The long arm is 32 by 10, and the standing arm adds 14 by 14 above it.

32 \times 10 + 14 \times 14 = 516

Each L face is 516 cm2.

2Trace the L's perimeter

#16 Count the Complement 6.G.A.4

Pushing each of the two step edges outward makes the outline a 32 by 24 rectangle, so the distance round is the same as that rectangle's.

(32 + 24) \times 2 = 112

112 cm round the L.

3Unroll the side faces

#17 Visualize Spatial Relationships 7.G.B.6

All eight of them together are one rectangle, 112 cm wide and 9 cm tall.

112 \times 9 = 1008

The sides are 1008 cm2 in total.

4Add the two L faces

#7 Identify Subproblems 7.G.B.6

The top and the bottom are congruent, so each contributes the same area.

516 \times 2 + 1008 = 2040

The surface area is 2040 cm2.

Answer: 2040 cm²

4 · Reviewdoes it hold up?

The two step edges are what makes it an L rather than a rectangle, and they face opposite ways -- so the outline really is 112 cm, the same as a 32 by 24 rectangle's.

Another way: Naming all eight side faces separately gives the same 1008 cm2, at the cost of eight chances to miss one.

Standardsmin grade 7

6.G.A.4 Surface area using nets of 3D figures — Finding the L face's area and the distance round it.
7.G.B.6 Solve real-world problems involving area, surface area, and volume — Getting a prism's surface as two bases plus a band.

💡Takeaway. The sides of any straight solid are one long strip. Its width is the distance round the base.

The sides of a prism are one band as wide as the base's perimeter · 12 practice problems

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