← A cut through a diagonal halves the solid without measuring the diagonal · Surface Area and Volume of Solids

A cut through a diagonal halves the solid without measuring the diagonal · 12 practice problems

6.G.A.27.G.A.3

Generated variants — 12

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

Variant 1 easy answer: 432 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 12 cm by 8 cm by 9 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 12 cm long, 8 cm deep and 9 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 12 cm by 8 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

12 \times 8 = 96,\quad 96 \div 2 = 48

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 9

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

12 \times 8 \times 9 \div 2 = 432

Each piece is 432 cm3.

Answer: 432 cm³

4 · Reviewdoes it hold up?

The two pieces together are 432 + 432 = 864 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 96 over 2, times height 9.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 2 easy answer: 504 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 14 cm by 9 cm by 8 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 14 cm long, 9 cm deep and 8 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 14 cm by 9 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

14 \times 9 = 126,\quad 126 \div 2 = 63

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 8

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

14 \times 9 \times 8 \div 2 = 504

Each piece is 504 cm3.

Answer: 504 cm³

4 · Reviewdoes it hold up?

The two pieces together are 504 + 504 = 1008 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 126 over 2, times height 8.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 3 easy answer: 900 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 15 cm by 12 cm by 10 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 15 cm long, 12 cm deep and 10 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 15 cm by 12 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

15 \times 12 = 180,\quad 180 \div 2 = 90

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 10

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

15 \times 12 \times 10 \div 2 = 900

Each piece is 900 cm3.

Answer: 900 cm³

4 · Reviewdoes it hold up?

The two pieces together are 900 + 900 = 1800 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 180 over 2, times height 10.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 4 easy answer: 560 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 16 cm by 10 cm by 7 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 16 cm long, 10 cm deep and 7 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 16 cm by 10 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

16 \times 10 = 160,\quad 160 \div 2 = 80

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 7

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

16 \times 10 \times 7 \div 2 = 560

Each piece is 560 cm3.

Answer: 560 cm³

4 · Reviewdoes it hold up?

The two pieces together are 560 + 560 = 1120 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 160 over 2, times height 7.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 5 medium answer: 594 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 18 cm by 6 cm by 11 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 18 cm long, 6 cm deep and 11 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 18 cm by 6 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

18 \times 6 = 108,\quad 108 \div 2 = 54

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 11

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

18 \times 6 \times 11 \div 2 = 594

Each piece is 594 cm3.

Answer: 594 cm³

4 · Reviewdoes it hold up?

The two pieces together are 594 + 594 = 1188 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 108 over 2, times height 11.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 6 medium answer: 700 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 20 cm by 14 cm by 5 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 20 cm long, 14 cm deep and 5 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 20 cm by 14 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

20 \times 14 = 280,\quad 280 \div 2 = 140

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 5

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

20 \times 14 \times 5 \div 2 = 700

Each piece is 700 cm3.

Answer: 700 cm³

4 · Reviewdoes it hold up?

The two pieces together are 700 + 700 = 1400 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 280 over 2, times height 5.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 7 medium answer: 660 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 22 cm by 10 cm by 6 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 22 cm long, 10 cm deep and 6 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 22 cm by 10 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

22 \times 10 = 220,\quad 220 \div 2 = 110

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 6

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

22 \times 10 \times 6 \div 2 = 660

Each piece is 660 cm3.

Answer: 660 cm³

4 · Reviewdoes it hold up?

The two pieces together are 660 + 660 = 1320 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 220 over 2, times height 6.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 8 medium answer: 720 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 24 cm by 15 cm by 4 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 24 cm long, 15 cm deep and 4 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 24 cm by 15 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

24 \times 15 = 360,\quad 360 \div 2 = 180

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 4

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

24 \times 15 \times 4 \div 2 = 720

Each piece is 720 cm3.

Answer: 720 cm³

4 · Reviewdoes it hold up?

The two pieces together are 720 + 720 = 1440 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 360 over 2, times height 4.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 9 hard answer: 1200 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 25 cm by 8 cm by 12 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 25 cm long, 8 cm deep and 12 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 25 cm by 8 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

25 \times 8 = 200,\quad 200 \div 2 = 100

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 12

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

25 \times 8 \times 12 \div 2 = 1200

Each piece is 1200 cm3.

Answer: 1200 cm³

4 · Reviewdoes it hold up?

The two pieces together are 1200 + 1200 = 2400 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 200 over 2, times height 12.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 10 hard answer: 1260 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 28 cm by 18 cm by 5 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 28 cm long, 18 cm deep and 5 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 28 cm by 18 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

28 \times 18 = 504,\quad 504 \div 2 = 252

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 5

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

28 \times 18 \times 5 \div 2 = 1260

Each piece is 1260 cm3.

Answer: 1260 cm³

4 · Reviewdoes it hold up?

The two pieces together are 1260 + 1260 = 2520 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 504 over 2, times height 5.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 11 hard answer: 1260 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 30 cm by 12 cm by 7 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 30 cm long, 12 cm deep and 7 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 30 cm by 12 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

30 \times 12 = 360,\quad 360 \div 2 = 180

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 7

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

30 \times 12 \times 7 \div 2 = 1260

Each piece is 1260 cm3.

Answer: 1260 cm³

4 · Reviewdoes it hold up?

The two pieces together are 1260 + 1260 = 2520 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 360 over 2, times height 7.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.

Variant 12 hard answer: 2880 cm³

The rectangular prism at the right is cut along the red line. What is the volume of one of the two pieces, in $\text{cm}^3$ ?

Show solution

1 · Understandwhat's really being asked

A prism 32 cm by 20 cm by 9 cm is cut straight down along a diagonal of its top face. We want the volume of one piece.

Givens

The prism is 32 cm long, 20 cm deep and 9 cm tall.
The cut follows a diagonal of the top face and goes straight down.

Unknowns

The volume of one of the two pieces.

Constraints

The cut is vertical, so it makes the same shape at every height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

The diagonal's length looks necessary and is not. A diagonal cuts a rectangle into two equal triangles, and a vertical cut carries that split all the way down -- so each piece is half the prism.

3 · Execute3 carry out the plan

1Look at the top face

#9 Solve an Easier Related Problem 7.G.A.3

It is a 32 cm by 20 cm rectangle, and a diagonal cuts any rectangle into two congruent triangles.

32 \times 20 = 640,\quad 640 \div 2 = 320

The base is halved.

2Carry the cut down

#17 Visualize Spatial Relationships 6.G.A.2

The cut is vertical, so every horizontal slice of the prism is split the same way.

\text{piece} = \text{half base} \times 9

Half at every height means half in total.

3Work out the volume

#7 Identify Subproblems 6.G.A.2

Half the whole prism.

32 \times 20 \times 9 \div 2 = 2880

Each piece is 2880 cm3.

Answer: 2880 cm³

4 · Reviewdoes it hold up?

The two pieces together are 2880 + 2880 = 5760 cm3, which is the whole prism -- nothing was lost in the cut.

Another way: Treating the piece as a prism with a triangular base gives the same: base 640 over 2, times height 9.

Standardsmin grade 7

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Describing the flat shape the cut leaves behind.
6.G.A.2 Volume of right rectangular prisms with fractional edges — Getting a prism's volume as base area times height.

💡Takeaway. If a cut goes straight down, whatever it does to the top face it does to the whole solid.