← Hidden edges are one of each length · Net and Solid Structure

Hidden edges are one of each length · 12 practice problems

5.MD.C.35.OA.A.1

Generated variants — 12

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

Variant 1 easy answer: 8 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 15 cm15 \text{ cm}. Find the value of ◯\bigcirc.

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

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 15 cm. Two of them are 3 cm and 4 cm, and we need the third.

Givens
  • The hidden edges total 15 cm.
  • Two of the three edge lengths are 3 cm and 4 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 15 cm.
◯+3+4=15\bigcirc + 3 + 4 = 15
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 3 cm and 4 cm, which total 7 cm.
3+4=73 + 4 = 7
7 cm of the 15 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 15 cm and the two known ones total 7 cm, the remaining edge is 15 - 7.
◯=15−7=8\bigcirc = 15 - 7 = 8
The circle stands for 8 cm.
Answer: 8 cm
4 · Reviewdoes it hold up?

Put it back: 8 + 3 + 4 = 15 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (8 + 3 + 4) = 60 cm, and the three hidden ones are a quarter of that -- 15 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 2 easy answer: 5 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 16 cm16 \text{ cm}. Find the value of ◯\bigcirc.

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

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 16 cm. Two of them are 4 cm and 7 cm, and we need the third.

Givens
  • The hidden edges total 16 cm.
  • Two of the three edge lengths are 4 cm and 7 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 16 cm.
◯+4+7=16\bigcirc + 4 + 7 = 16
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 4 cm and 7 cm, which total 11 cm.
4+7=114 + 7 = 11
11 cm of the 16 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 16 cm and the two known ones total 11 cm, the remaining edge is 16 - 11.
◯=16−11=5\bigcirc = 16 - 11 = 5
The circle stands for 5 cm.
Answer: 5 cm
4 · Reviewdoes it hold up?

Put it back: 5 + 4 + 7 = 16 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (5 + 4 + 7) = 64 cm, and the three hidden ones are a quarter of that -- 16 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 3 easy answer: 8 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 18 cm18 \text{ cm}. Find the value of ◯\bigcirc.

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

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 18 cm. Two of them are 5 cm and 5 cm, and we need the third.

Givens
  • The hidden edges total 18 cm.
  • Two of the three edge lengths are 5 cm and 5 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 18 cm.
◯+5+5=18\bigcirc + 5 + 5 = 18
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 5 cm and 5 cm, which total 10 cm.
5+5=105 + 5 = 10
10 cm of the 18 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 18 cm and the two known ones total 10 cm, the remaining edge is 18 - 10.
◯=18−10=8\bigcirc = 18 - 10 = 8
The circle stands for 8 cm.
Answer: 8 cm
4 · Reviewdoes it hold up?

Put it back: 8 + 5 + 5 = 18 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (8 + 5 + 5) = 72 cm, and the three hidden ones are a quarter of that -- 18 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 4 easy answer: 6 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 19 cm19 \text{ cm}. Find the value of ◯\bigcirc.

2 cm 11 cm ○ cm
Show solution
1 · Understandwhat's really being asked

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 19 cm. Two of them are 2 cm and 11 cm, and we need the third.

Givens
  • The hidden edges total 19 cm.
  • Two of the three edge lengths are 2 cm and 11 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 19 cm.
◯+2+11=19\bigcirc + 2 + 11 = 19
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 2 cm and 11 cm, which total 13 cm.
2+11=132 + 11 = 13
13 cm of the 19 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 19 cm and the two known ones total 13 cm, the remaining edge is 19 - 13.
◯=19−13=6\bigcirc = 19 - 13 = 6
The circle stands for 6 cm.
Answer: 6 cm
4 · Reviewdoes it hold up?

Put it back: 6 + 2 + 11 = 19 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (6 + 2 + 11) = 76 cm, and the three hidden ones are a quarter of that -- 19 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 5 medium answer: 8 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 20 cm20 \text{ cm}. Find the value of ◯\bigcirc.

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

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 20 cm. Two of them are 3 cm and 9 cm, and we need the third.

Givens
  • The hidden edges total 20 cm.
  • Two of the three edge lengths are 3 cm and 9 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 20 cm.
◯+3+9=20\bigcirc + 3 + 9 = 20
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 3 cm and 9 cm, which total 12 cm.
3+9=123 + 9 = 12
12 cm of the 20 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 20 cm and the two known ones total 12 cm, the remaining edge is 20 - 12.
◯=20−12=8\bigcirc = 20 - 12 = 8
The circle stands for 8 cm.
Answer: 8 cm
4 · Reviewdoes it hold up?

Put it back: 8 + 3 + 9 = 20 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (8 + 3 + 9) = 80 cm, and the three hidden ones are a quarter of that -- 20 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 6 medium answer: 7 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 21 cm21 \text{ cm}. Find the value of ◯\bigcirc.

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

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 21 cm. Two of them are 6 cm and 8 cm, and we need the third.

Givens
  • The hidden edges total 21 cm.
  • Two of the three edge lengths are 6 cm and 8 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 21 cm.
◯+6+8=21\bigcirc + 6 + 8 = 21
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 6 cm and 8 cm, which total 14 cm.
6+8=146 + 8 = 14
14 cm of the 21 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 21 cm and the two known ones total 14 cm, the remaining edge is 21 - 14.
◯=21−14=7\bigcirc = 21 - 14 = 7
The circle stands for 7 cm.
Answer: 7 cm
4 · Reviewdoes it hold up?

Put it back: 7 + 6 + 8 = 21 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (7 + 6 + 8) = 84 cm, and the three hidden ones are a quarter of that -- 21 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 7 medium answer: 5 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 24 cm24 \text{ cm}. Find the value of ◯\bigcirc.

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

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 24 cm. Two of them are 9 cm and 10 cm, and we need the third.

Givens
  • The hidden edges total 24 cm.
  • Two of the three edge lengths are 9 cm and 10 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 24 cm.
◯+9+10=24\bigcirc + 9 + 10 = 24
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 9 cm and 10 cm, which total 19 cm.
9+10=199 + 10 = 19
19 cm of the 24 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 24 cm and the two known ones total 19 cm, the remaining edge is 24 - 19.
◯=24−19=5\bigcirc = 24 - 19 = 5
The circle stands for 5 cm.
Answer: 5 cm
4 · Reviewdoes it hold up?

Put it back: 5 + 9 + 10 = 24 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (5 + 9 + 10) = 96 cm, and the three hidden ones are a quarter of that -- 24 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 8 medium answer: 6 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 25 cm25 \text{ cm}. Find the value of ◯\bigcirc.

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

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 25 cm. Two of them are 7 cm and 12 cm, and we need the third.

Givens
  • The hidden edges total 25 cm.
  • Two of the three edge lengths are 7 cm and 12 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 25 cm.
◯+7+12=25\bigcirc + 7 + 12 = 25
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 7 cm and 12 cm, which total 19 cm.
7+12=197 + 12 = 19
19 cm of the 25 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 25 cm and the two known ones total 19 cm, the remaining edge is 25 - 19.
◯=25−19=6\bigcirc = 25 - 19 = 6
The circle stands for 6 cm.
Answer: 6 cm
4 · Reviewdoes it hold up?

Put it back: 6 + 7 + 12 = 25 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (6 + 7 + 12) = 100 cm, and the three hidden ones are a quarter of that -- 25 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 9 hard answer: 8 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 26 cm26 \text{ cm}. Find the value of ◯\bigcirc.

4 cm 14 cm ○ cm
Show solution
1 · Understandwhat's really being asked

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 26 cm. Two of them are 4 cm and 14 cm, and we need the third.

Givens
  • The hidden edges total 26 cm.
  • Two of the three edge lengths are 4 cm and 14 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 26 cm.
◯+4+14=26\bigcirc + 4 + 14 = 26
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 4 cm and 14 cm, which total 18 cm.
4+14=184 + 14 = 18
18 cm of the 26 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 26 cm and the two known ones total 18 cm, the remaining edge is 26 - 18.
◯=26−18=8\bigcirc = 26 - 18 = 8
The circle stands for 8 cm.
Answer: 8 cm
4 · Reviewdoes it hold up?

Put it back: 8 + 4 + 14 = 26 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (8 + 4 + 14) = 104 cm, and the three hidden ones are a quarter of that -- 26 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 10 hard answer: 9 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 28 cm28 \text{ cm}. Find the value of ◯\bigcirc.

6 cm 13 cm ○ cm
Show solution
1 · Understandwhat's really being asked

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 28 cm. Two of them are 6 cm and 13 cm, and we need the third.

Givens
  • The hidden edges total 28 cm.
  • Two of the three edge lengths are 6 cm and 13 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 28 cm.
◯+6+13=28\bigcirc + 6 + 13 = 28
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 6 cm and 13 cm, which total 19 cm.
6+13=196 + 13 = 19
19 cm of the 28 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 28 cm and the two known ones total 19 cm, the remaining edge is 28 - 19.
◯=28−19=9\bigcirc = 28 - 19 = 9
The circle stands for 9 cm.
Answer: 9 cm
4 · Reviewdoes it hold up?

Put it back: 9 + 6 + 13 = 28 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (9 + 6 + 13) = 112 cm, and the three hidden ones are a quarter of that -- 28 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 11 hard answer: 7 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 30 cm30 \text{ cm}. Find the value of ◯\bigcirc.

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

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 30 cm. Two of them are 8 cm and 15 cm, and we need the third.

Givens
  • The hidden edges total 30 cm.
  • Two of the three edge lengths are 8 cm and 15 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 30 cm.
◯+8+15=30\bigcirc + 8 + 15 = 30
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 8 cm and 15 cm, which total 23 cm.
8+15=238 + 15 = 23
23 cm of the 30 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 30 cm and the two known ones total 23 cm, the remaining edge is 30 - 23.
◯=30−23=7\bigcirc = 30 - 23 = 7
The circle stands for 7 cm.
Answer: 7 cm
4 · Reviewdoes it hold up?

Put it back: 7 + 8 + 15 = 30 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (7 + 8 + 15) = 120 cm, and the three hidden ones are a quarter of that -- 30 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.
Variant 12 hard answer: 9 cm

In the sketch (cabinet drawing) of the rectangular prism on the right, the sum of the lengths of the hidden edges is 32 cm32 \text{ cm}. Find the value of ◯\bigcirc.

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

A rectangular prism is sketched with its hidden edges dashed. Those three dashed edges add to 32 cm. Two of them are 11 cm and 12 cm, and we need the third.

Givens
  • The hidden edges total 32 cm.
  • Two of the three edge lengths are 11 cm and 12 cm.
  • Exactly one hidden edge has each of the three lengths.
Unknowns
  • The length marked with the circle.
Constraints
  • A rectangular prism has three edge lengths, and exactly one edge of each is hidden in a cabinet sketch.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Work out which edges the dashed lines are first, because that decides what goes into the sum. Once the sum is written down, the missing piece comes out by subtraction.

3 · Execute4 carry out the plan

1See which edges are hidden

#17 Visualize Spatial Relationships 5.MD.C.3
Picture the rectangular prism. The edges drawn as dashed lines are the ones tucked behind the box: they all meet at the one corner you cannot see. There are three of them, one of each length, not three of the same length.
One width, one depth, one height -- hidden.

2Write the total of the hidden edges

#1 Draw a Diagram 5.OA.A.1
Add up the three hidden edge lengths and set the sum equal to the given 32 cm.
◯+11+12=32\bigcirc + 11 + 12 = 32
One equation, one unknown.

3Combine the known parts

#1 Draw a Diagram 5.OA.A.1
The two known hidden edges are 11 cm and 12 cm, which total 23 cm.
11+12=2311 + 12 = 23
23 cm of the 32 cm is accounted for.

4Work backwards to find the unknown edge

#11 Work Backwards 5.OA.A.1
Since the three hidden edges total 32 cm and the two known ones total 23 cm, the remaining edge is 32 - 23.
◯=32−23=9\bigcirc = 32 - 23 = 9
The circle stands for 9 cm.
Answer: 9 cm
4 · Reviewdoes it hold up?

Put it back: 9 + 11 + 12 = 32 cm, which is the sum the problem gave.

Another way: A check on the whole solid: all twelve edges together are 4 x (9 + 11 + 12) = 128 cm, and the three hidden ones are a quarter of that -- 32 cm, as stated.

Standardsmin grade 5
  • 5.MD.C.3 Recognize volume as an attribute of solid figures — Reading a cabinet sketch as a solid with three edge lengths.
  • 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions — Writing and unpicking the sum of the three hidden edges.
💡Takeaway. In a sketch of a box exactly three edges hide, and they are one of each length -- so their total is one of each, added up.