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Four side faces in a row are one rectangle as wide as the base's perimeter · 12 practice problems

6.G.A.46.G.A.1

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Generated variants — 12

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

Variant 1 easy

In the net of the rectangular prism at the right, rectangle ABCD has an area of $96\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 96 cm2. The base is 10 cm by 6 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 96 cm2.
The base measures 10 cm by 6 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 10, 6, 10 and 6.

(10 + 6) \times 2 = 32

The strip is 32 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

32 \times \text{AB} = 96

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

96 \div 32 = 3

AB is 3 cm.

Answer: 3 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 10 by 6 by 3 cm, whose four side faces are 10x3 and 6x3 twice over -- and those add to 96 cm2.

Another way: Adding the four faces separately gives $2 \times 10 \times \text{AB} + 2 \times 6 \times \text{AB} = 96$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 2 easy

In the net of the rectangular prism at the right, rectangle ABCD has an area of $104\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 104 cm2. The base is 8 cm by 5 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 104 cm2.
The base measures 8 cm by 5 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 8, 5, 8 and 5.

(8 + 5) \times 2 = 26

The strip is 26 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

26 \times \text{AB} = 104

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

104 \div 26 = 4

AB is 4 cm.

Answer: 4 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 8 by 5 by 4 cm, whose four side faces are 8x4 and 5x4 twice over -- and those add to 104 cm2.

Another way: Adding the four faces separately gives $2 \times 8 \times \text{AB} + 2 \times 5 \times \text{AB} = 104$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 3 easy

In the net of the rectangular prism at the right, rectangle ABCD has an area of $126\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 126 cm2. The base is 5 cm by 2 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 126 cm2.
The base measures 5 cm by 2 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 5, 2, 5 and 2.

(5 + 2) \times 2 = 14

The strip is 14 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

14 \times \text{AB} = 126

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

126 \div 14 = 9

AB is 9 cm.

Answer: 9 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 5 by 2 by 9 cm, whose four side faces are 5x9 and 2x9 twice over -- and those add to 126 cm2.

Another way: Adding the four faces separately gives $2 \times 5 \times \text{AB} + 2 \times 2 \times \text{AB} = 126$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 4 easy

In the net of the rectangular prism at the right, rectangle ABCD has an area of $132\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 132 cm2. The base is 7 cm by 4 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 132 cm2.
The base measures 7 cm by 4 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 7, 4, 7 and 4.

(7 + 4) \times 2 = 22

The strip is 22 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

22 \times \text{AB} = 132

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

132 \div 22 = 6

AB is 6 cm.

Answer: 6 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 7 by 4 by 6 cm, whose four side faces are 7x6 and 4x6 twice over -- and those add to 132 cm2.

Another way: Adding the four faces separately gives $2 \times 7 \times \text{AB} + 2 \times 4 \times \text{AB} = 132$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 5 medium

In the net of the rectangular prism at the right, rectangle ABCD has an area of $144\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 144 cm2. The base is 6 cm by 3 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 144 cm2.
The base measures 6 cm by 3 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 6, 3, 6 and 3.

(6 + 3) \times 2 = 18

The strip is 18 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

18 \times \text{AB} = 144

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

144 \div 18 = 8

AB is 8 cm.

Answer: 8 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 6 by 3 by 8 cm, whose four side faces are 6x8 and 3x8 twice over -- and those add to 144 cm2.

Another way: Adding the four faces separately gives $2 \times 6 \times \text{AB} + 2 \times 3 \times \text{AB} = 144$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 6 medium

In the net of the rectangular prism at the right, rectangle ABCD has an area of $150\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 150 cm2. The base is 9 cm by 6 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 150 cm2.
The base measures 9 cm by 6 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 9, 6, 9 and 6.

(9 + 6) \times 2 = 30

The strip is 30 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

30 \times \text{AB} = 150

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

150 \div 30 = 5

AB is 5 cm.

Answer: 5 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 9 by 6 by 5 cm, whose four side faces are 9x5 and 6x5 twice over -- and those add to 150 cm2.

Another way: Adding the four faces separately gives $2 \times 9 \times \text{AB} + 2 \times 6 \times \text{AB} = 150$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 7 medium

In the net of the rectangular prism at the right, rectangle ABCD has an area of $152\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 152 cm2. The base is 11 cm by 8 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 152 cm2.
The base measures 11 cm by 8 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 11, 8, 11 and 8.

(11 + 8) \times 2 = 38

The strip is 38 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

38 \times \text{AB} = 152

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

152 \div 38 = 4

AB is 4 cm.

Answer: 4 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 11 by 8 by 4 cm, whose four side faces are 11x4 and 8x4 twice over -- and those add to 152 cm2.

Another way: Adding the four faces separately gives $2 \times 11 \times \text{AB} + 2 \times 8 \times \text{AB} = 152$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 8 medium

In the net of the rectangular prism at the right, rectangle ABCD has an area of $156\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 156 cm2. The base is 15 cm by 11 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 156 cm2.
The base measures 15 cm by 11 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 15, 11, 15 and 11.

(15 + 11) \times 2 = 52

The strip is 52 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

52 \times \text{AB} = 156

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

156 \div 52 = 3

AB is 3 cm.

Answer: 3 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 15 by 11 by 3 cm, whose four side faces are 15x3 and 11x3 twice over -- and those add to 156 cm2.

Another way: Adding the four faces separately gives $2 \times 15 \times \text{AB} + 2 \times 11 \times \text{AB} = 156$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 9 hard

In the net of the rectangular prism at the right, rectangle ABCD has an area of $190\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 190 cm2. The base is 12 cm by 7 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 190 cm2.
The base measures 12 cm by 7 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 12, 7, 12 and 7.

(12 + 7) \times 2 = 38

The strip is 38 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

38 \times \text{AB} = 190

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

190 \div 38 = 5

AB is 5 cm.

Answer: 5 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 12 by 7 by 5 cm, whose four side faces are 12x5 and 7x5 twice over -- and those add to 190 cm2.

Another way: Adding the four faces separately gives $2 \times 12 \times \text{AB} + 2 \times 7 \times \text{AB} = 190$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 10 hard

In the net of the rectangular prism at the right, rectangle ABCD has an area of $276\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 276 cm2. The base is 14 cm by 9 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 276 cm2.
The base measures 14 cm by 9 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 14, 9, 14 and 9.

(14 + 9) \times 2 = 46

The strip is 46 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

46 \times \text{AB} = 276

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

276 \div 46 = 6

AB is 6 cm.

Answer: 6 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 14 by 9 by 6 cm, whose four side faces are 14x6 and 9x6 twice over -- and those add to 276 cm2.

Another way: Adding the four faces separately gives $2 \times 14 \times \text{AB} + 2 \times 9 \times \text{AB} = 276$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 11 hard

In the net of the rectangular prism at the right, rectangle ABCD has an area of $286\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 286 cm2. The base is 9 cm by 4 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 286 cm2.
The base measures 9 cm by 4 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 9, 4, 9 and 4.

(9 + 4) \times 2 = 26

The strip is 26 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

26 \times \text{AB} = 286

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

286 \div 26 = 11

AB is 11 cm.

Answer: 11 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 9 by 4 by 11 cm, whose four side faces are 9x11 and 4x11 twice over -- and those add to 286 cm2.

Another way: Adding the four faces separately gives $2 \times 9 \times \text{AB} + 2 \times 4 \times \text{AB} = 286$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Variant 12 hard

In the net of the rectangular prism at the right, rectangle ABCD has an area of $322\ \text{cm}^2$ . How long is segment AB, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

The four side faces of a rectangular prism lie in a row and make one rectangle ABCD of area 322 cm2. The base is 13 cm by 10 cm, and we want the length of AB.

Givens

Rectangle ABCD has an area of 322 cm2.
The base measures 13 cm by 10 cm.
The four side faces sit side by side with no gaps.

Unknowns

The length of segment AB.

Constraints

AB is a side of the strip, so it is the prism's height.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#1 Draw a Diagram

Treat the four side faces as one rectangle. Its long side is what the prism's base measures all the way round, so the area gives the other side in one division.

3 · Execute3 carry out the plan

1Find how wide the strip is

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

Laid end to end the four faces span the base's whole perimeter: 13, 10, 13 and 10.

(13 + 10) \times 2 = 46

The strip is 46 cm long.

2Use the area of ABCD

#1 Draw a Diagram 6.G.A.1

The strip is a rectangle, so its area is the two sides multiplied.

46 \times \text{AB} = 322

One side known, one to find.

3Divide to get AB

#7 Identify Subproblems 6.G.A.1

Undo the multiplication.

322 \div 46 = 7

AB is 7 cm.

Answer: 7 cm

4 · Reviewdoes it hold up?

Folding it back up gives a prism 13 by 10 by 7 cm, whose four side faces are 13x7 and 10x7 twice over -- and those add to 322 cm2.

Another way: Adding the four faces separately gives $2 \times 13 \times \text{AB} + 2 \times 10 \times \text{AB} = 322$ , the same equation with the factoring left undone.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Seeing the four side faces of the net as one rectangle.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Getting the missing side of a rectangle from its area.

💡Takeaway. The side faces of a box, unrolled, make one long rectangle as wide as the way round the base.

Where next?

← Back to the problem: Four side faces in a row are one rectangle as wide as the base's perimeter