← Each band is a perimeter, and its direction says which one · Net and Solid Structure

Each band is a perimeter, and its direction says which one · 12 practice problems

6.G.A.44.MD.A.3

Generated variants — 12

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

Variant 1 easy answer: 560 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

60 cm 20 cm 30 cm
Show solution
1 · Understandwhat's really being asked

A box 60 cm by 30 cm by 20 cm is tied with 4 bands one way and 1 the other. We want the total length of ribbon.

Givens
  • The box is 60 cm long, 30 cm deep and 20 cm tall.
  • 4 bands run the short way round.
  • 1 band run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 30 cm depth twice and climbs the 20 cm height twice.
(30+20)×2=100(30 + 20) \times 2 = 100
Each of those bands is 100 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 60 cm length twice and the 20 cm height twice.
(60+20)×2=160(60 + 20) \times 2 = 160
Each of those is 160 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 4 the short way and 1 the long way.
100×4=400,160×1=160100 \times 4 = 400,\quad 160 \times 1 = 160
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
400+160=560400 + 160 = 560
560 cm of ribbon.
Answer: 560 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 10 times -- 200 cm of the total -- and the rest, 360 cm, lies along the top and bottom. The two add back to 560.

Another way: Counting the ribbon face by face gives the same total, but it needs 20 separate pieces instead of 5 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 2 easy answer: 1110 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

70 cm 35 cm 45 cm
Show solution
1 · Understandwhat's really being asked

A box 70 cm by 45 cm by 35 cm is tied with 3 bands one way and 3 the other. We want the total length of ribbon.

Givens
  • The box is 70 cm long, 45 cm deep and 35 cm tall.
  • 3 bands run the short way round.
  • 3 bands run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 45 cm depth twice and climbs the 35 cm height twice.
(45+35)×2=160(45 + 35) \times 2 = 160
Each of those bands is 160 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 70 cm length twice and the 35 cm height twice.
(70+35)×2=210(70 + 35) \times 2 = 210
Each of those is 210 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 3 the short way and 3 the long way.
160×3=480,210×3=630160 \times 3 = 480,\quad 210 \times 3 = 630
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
480+630=1110480 + 630 = 1110
1110 cm of ribbon.
Answer: 1110 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 12 times -- 420 cm of the total -- and the rest, 690 cm, lies along the top and bottom. The two add back to 1110.

Another way: Counting the ribbon face by face gives the same total, but it needs 24 separate pieces instead of 6 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 3 easy answer: 700 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

80 cm 50 cm 60 cm
Show solution
1 · Understandwhat's really being asked

A box 80 cm by 60 cm by 50 cm is tied with 2 bands one way and 1 the other. We want the total length of ribbon.

Givens
  • The box is 80 cm long, 60 cm deep and 50 cm tall.
  • 2 bands run the short way round.
  • 1 band run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 60 cm depth twice and climbs the 50 cm height twice.
(60+50)×2=220(60 + 50) \times 2 = 220
Each of those bands is 220 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 80 cm length twice and the 50 cm height twice.
(80+50)×2=260(80 + 50) \times 2 = 260
Each of those is 260 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 2 the short way and 1 the long way.
220×2=440,260×1=260220 \times 2 = 440,\quad 260 \times 1 = 260
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
440+260=700440 + 260 = 700
700 cm of ribbon.
Answer: 700 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 6 times -- 300 cm of the total -- and the rest, 400 cm, lies along the top and bottom. The two add back to 700.

Another way: Counting the ribbon face by face gives the same total, but it needs 12 separate pieces instead of 3 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 4 easy answer: 700 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

90 cm 40 cm 50 cm
Show solution
1 · Understandwhat's really being asked

A box 90 cm by 50 cm by 40 cm is tied with 1 band one way and 2 the other. We want the total length of ribbon.

Givens
  • The box is 90 cm long, 50 cm deep and 40 cm tall.
  • 1 band run the short way round.
  • 2 bands run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 50 cm depth twice and climbs the 40 cm height twice.
(50+40)×2=180(50 + 40) \times 2 = 180
Each of those bands is 180 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 90 cm length twice and the 40 cm height twice.
(90+40)×2=260(90 + 40) \times 2 = 260
Each of those is 260 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 1 the short way and 2 the long way.
180×1=180,260×2=520180 \times 1 = 180,\quad 260 \times 2 = 520
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
180+520=700180 + 520 = 700
700 cm of ribbon.
Answer: 700 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 6 times -- 240 cm of the total -- and the rest, 460 cm, lies along the top and bottom. The two add back to 700.

Another way: Counting the ribbon face by face gives the same total, but it needs 12 separate pieces instead of 3 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 5 medium answer: 500 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

95 cm 45 cm 65 cm
Show solution
1 · Understandwhat's really being asked

A box 95 cm by 65 cm by 45 cm is tied with 1 band one way and 1 the other. We want the total length of ribbon.

Givens
  • The box is 95 cm long, 65 cm deep and 45 cm tall.
  • 1 band run the short way round.
  • 1 band run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 65 cm depth twice and climbs the 45 cm height twice.
(65+45)×2=220(65 + 45) \times 2 = 220
Each of those bands is 220 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 95 cm length twice and the 45 cm height twice.
(95+45)×2=280(95 + 45) \times 2 = 280
Each of those is 280 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 1 the short way and 1 the long way.
220×1=220,280×1=280220 \times 1 = 220,\quad 280 \times 1 = 280
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
220+280=500220 + 280 = 500
500 cm of ribbon.
Answer: 500 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 4 times -- 180 cm of the total -- and the rest, 320 cm, lies along the top and bottom. The two add back to 500.

Another way: Counting the ribbon face by face gives the same total, but it needs 8 separate pieces instead of 2 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 6 medium answer: 940 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

100 cm 30 cm 40 cm
Show solution
1 · Understandwhat's really being asked

A box 100 cm by 40 cm by 30 cm is tied with 3 bands one way and 2 the other. We want the total length of ribbon.

Givens
  • The box is 100 cm long, 40 cm deep and 30 cm tall.
  • 3 bands run the short way round.
  • 2 bands run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 40 cm depth twice and climbs the 30 cm height twice.
(40+30)×2=140(40 + 30) \times 2 = 140
Each of those bands is 140 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 100 cm length twice and the 30 cm height twice.
(100+30)×2=260(100 + 30) \times 2 = 260
Each of those is 260 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 3 the short way and 2 the long way.
140×3=420,260×2=520140 \times 3 = 420,\quad 260 \times 2 = 520
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
420+520=940420 + 520 = 940
940 cm of ribbon.
Answer: 940 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 10 times -- 300 cm of the total -- and the rest, 640 cm, lies along the top and bottom. The two add back to 940.

Another way: Counting the ribbon face by face gives the same total, but it needs 20 separate pieces instead of 5 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 7 medium answer: 1400 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

110 cm 80 cm 90 cm
Show solution
1 · Understandwhat's really being asked

A box 110 cm by 90 cm by 80 cm is tied with 3 bands one way and 1 the other. We want the total length of ribbon.

Givens
  • The box is 110 cm long, 90 cm deep and 80 cm tall.
  • 3 bands run the short way round.
  • 1 band run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 90 cm depth twice and climbs the 80 cm height twice.
(90+80)×2=340(90 + 80) \times 2 = 340
Each of those bands is 340 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 110 cm length twice and the 80 cm height twice.
(110+80)×2=380(110 + 80) \times 2 = 380
Each of those is 380 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 3 the short way and 1 the long way.
340×3=1020,380×1=380340 \times 3 = 1020,\quad 380 \times 1 = 380
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
1020+380=14001020 + 380 = 1400
1400 cm of ribbon.
Answer: 1400 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 8 times -- 640 cm of the total -- and the rest, 760 cm, lies along the top and bottom. The two add back to 1400.

Another way: Counting the ribbon face by face gives the same total, but it needs 16 separate pieces instead of 4 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 8 medium answer: 1240 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

120 cm 60 cm 70 cm
Show solution
1 · Understandwhat's really being asked

A box 120 cm by 70 cm by 60 cm is tied with 2 bands one way and 2 the other. We want the total length of ribbon.

Givens
  • The box is 120 cm long, 70 cm deep and 60 cm tall.
  • 2 bands run the short way round.
  • 2 bands run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 70 cm depth twice and climbs the 60 cm height twice.
(70+60)×2=260(70 + 60) \times 2 = 260
Each of those bands is 260 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 120 cm length twice and the 60 cm height twice.
(120+60)×2=360(120 + 60) \times 2 = 360
Each of those is 360 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 2 the short way and 2 the long way.
260×2=520,360×2=720260 \times 2 = 520,\quad 360 \times 2 = 720
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
520+720=1240520 + 720 = 1240
1240 cm of ribbon.
Answer: 1240 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 8 times -- 480 cm of the total -- and the rest, 760 cm, lies along the top and bottom. The two add back to 1240.

Another way: Counting the ribbon face by face gives the same total, but it needs 16 separate pieces instead of 4 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 9 hard answer: 830 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

130 cm 55 cm 60 cm
Show solution
1 · Understandwhat's really being asked

A box 130 cm by 60 cm by 55 cm is tied with 2 bands one way and 1 the other. We want the total length of ribbon.

Givens
  • The box is 130 cm long, 60 cm deep and 55 cm tall.
  • 2 bands run the short way round.
  • 1 band run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 60 cm depth twice and climbs the 55 cm height twice.
(60+55)×2=230(60 + 55) \times 2 = 230
Each of those bands is 230 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 130 cm length twice and the 55 cm height twice.
(130+55)×2=370(130 + 55) \times 2 = 370
Each of those is 370 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 2 the short way and 1 the long way.
230×2=460,370×1=370230 \times 2 = 460,\quad 370 \times 1 = 370
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
460+370=830460 + 370 = 830
830 cm of ribbon.
Answer: 830 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 6 times -- 330 cm of the total -- and the rest, 500 cm, lies along the top and bottom. The two add back to 830.

Another way: Counting the ribbon face by face gives the same total, but it needs 12 separate pieces instead of 3 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 10 hard answer: 1940 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

140 cm 70 cm 100 cm
Show solution
1 · Understandwhat's really being asked

A box 140 cm by 100 cm by 70 cm is tied with 2 bands one way and 3 the other. We want the total length of ribbon.

Givens
  • The box is 140 cm long, 100 cm deep and 70 cm tall.
  • 2 bands run the short way round.
  • 3 bands run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 100 cm depth twice and climbs the 70 cm height twice.
(100+70)×2=340(100 + 70) \times 2 = 340
Each of those bands is 340 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 140 cm length twice and the 70 cm height twice.
(140+70)×2=420(140 + 70) \times 2 = 420
Each of those is 420 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 2 the short way and 3 the long way.
340×2=680,420×3=1260340 \times 2 = 680,\quad 420 \times 3 = 1260
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
680+1260=1940680 + 1260 = 1940
1940 cm of ribbon.
Answer: 1940 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 10 times -- 700 cm of the total -- and the rest, 1240 cm, lies along the top and bottom. The two add back to 1940.

Another way: Counting the ribbon face by face gives the same total, but it needs 20 separate pieces instead of 5 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 11 hard answer: 1780 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

150 cm 90 cm 80 cm
Show solution
1 · Understandwhat's really being asked

A box 150 cm by 80 cm by 90 cm is tied with 1 band one way and 3 the other. We want the total length of ribbon.

Givens
  • The box is 150 cm long, 80 cm deep and 90 cm tall.
  • 1 band run the short way round.
  • 3 bands run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 80 cm depth twice and climbs the 90 cm height twice.
(80+90)×2=340(80 + 90) \times 2 = 340
Each of those bands is 340 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 150 cm length twice and the 90 cm height twice.
(150+90)×2=480(150 + 90) \times 2 = 480
Each of those is 480 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 1 the short way and 3 the long way.
340×1=340,480×3=1440340 \times 1 = 340,\quad 480 \times 3 = 1440
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
340+1440=1780340 + 1440 = 1780
1780 cm of ribbon.
Answer: 1780 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 8 times -- 720 cm of the total -- and the rest, 1060 cm, lies along the top and bottom. The two add back to 1780.

Another way: Counting the ribbon face by face gives the same total, but it needs 16 separate pieces instead of 4 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.
Variant 12 hard answer: 2050 cm

A box shaped like a rectangular prism is tied with ribbon as shown at the right. How many centimeters of ribbon are needed? (Ignore the knot.)

160 cm 75 cm 110 cm
Show solution
1 · Understandwhat's really being asked

A box 160 cm by 110 cm by 75 cm is tied with 3 bands one way and 2 the other. We want the total length of ribbon.

Givens
  • The box is 160 cm long, 110 cm deep and 75 cm tall.
  • 3 bands run the short way round.
  • 2 bands run the long way round.
Unknowns
  • The total centimetres of ribbon.
Constraints
  • The ribbon lies flat against the box and the knot is ignored.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #2 Make a Systematic List#1 Draw a Diagram

Each band goes all the way round the box, so its length is a perimeter -- but the two directions wrap different pairs of edges. Sort the bands by direction, then it is one multiplication each.

3 · Execute4 carry out the plan

1A band the short way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 110 cm depth twice and climbs the 75 cm height twice.
(110+75)×2=370(110 + 75) \times 2 = 370
Each of those bands is 370 cm.

2A band the long way round

#17 Visualize Spatial Relationships 6.G.A.4
It crosses the 160 cm length twice and the 75 cm height twice.
(160+75)×2=470(160 + 75) \times 2 = 470
Each of those is 470 cm.

3Count the bands of each kind

#2 Make a Systematic List 4.MD.A.3
There are 3 the short way and 2 the long way.
370×3=1110,470×2=940370 \times 3 = 1110,\quad 470 \times 2 = 940
Two groups, counted separately.

4Add the two groups

#2 Make a Systematic List 4.MD.A.3
The bands do not share any ribbon.
1110+940=20501110 + 940 = 2050
2050 cm of ribbon.
Answer: 2050 cm
4 · Reviewdoes it hold up?

Every band climbs the height twice, so the height is used 10 times -- 750 cm of the total -- and the rest, 1300 cm, lies along the top and bottom. The two add back to 2050.

Another way: Counting the ribbon face by face gives the same total, but it needs 20 separate pieces instead of 5 loops.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Seeing each band as a rectangle made of four of the box's edges.
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the perimeter formula and totalling the bands.
💡Takeaway. A ribbon round a box is a perimeter. Which perimeter depends on which way it goes.