← The sides of a prism are one band as wide as the base's perimeter · Surface Area and Volume of Solids

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

6.G.A.47.G.B.6

Generated variants — 12

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

Variant 1 easy answer: 856 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

14 cm 10 cm 4 cm 6 cm 14 cm
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.
14×4+6×6=9214 \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)×2=48(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.
48×14=67248 \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.
92×2+672=85692 \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 answer: 726 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

15 cm 9 cm 3 cm 5 cm 12 cm
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.
15×3+6×5=7515 \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)×2=48(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.
48×12=57648 \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.
75×2+576=72675 \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 answer: 1180 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

16 cm 11 cm 5 cm 4 cm 18 cm
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.
16×5+6×4=10416 \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)×2=54(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.
54×18=97254 \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.
104×2+972=1180104 \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 answer: 1232 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

20 cm 12 cm 4 cm 7 cm 15 cm
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.
20×4+8×7=13620 \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)×2=64(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.
64×15=96064 \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.
136×2+960=1232136 \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 answer: 1360 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

18 cm 10 cm 5 cm 6 cm 20 cm
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.
18×5+5×6=12018 \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)×2=56(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.
56×20=112056 \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.
120×2+1120=1360120 \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 answer: 1610 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

22 cm 15 cm 6 cm 9 cm 16 cm
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.
22×6+9×9=21322 \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)×2=74(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.
74×16=118474 \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.
213×2+1184=1610213 \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 answer: 2316 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

24 cm 14 cm 6 cm 8 cm 25 cm
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.
24×6+8×8=20824 \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)×2=76(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.
76×25=190076 \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.
208×2+1900=2316208 \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 answer: 2544 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

28 cm 16 cm 7 cm 12 cm 22 cm
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.
28×7+9×12=30428 \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)×2=88(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.
88×22=193688 \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.
304×2+1936=2544304 \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 answer: 3286 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

26 cm 20 cm 9 cm 11 cm 28 cm
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.
26×9+11×11=35526 \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)×2=92(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.
92×28=257692 \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.
355×2+2576=3286355 \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 answer: 1312 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

12 cm 8 cm 3 cm 4 cm 30 cm
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.
12×3+5×4=5612 \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)×2=40(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.
40×30=120040 \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.
56×2+1200=131256 \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 answer: 1640 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

30 cm 18 cm 8 cm 10 cm 10 cm
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.
30×8+10×10=34030 \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)×2=96(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.
96×10=96096 \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.
340×2+960=1640340 \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 answer: 2040 cm²

In the solid at the right every pair of meeting edges is perpendicular. What is its surface area, in cm2\text{cm}^2?

32 cm 24 cm 10 cm 14 cm 9 cm
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.
32×10+14×14=51632 \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)×2=112(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.
112×9=1008112 \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.
516×2+1008=2040516 \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.