← 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

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Generated variants — 12

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

Variant 1 easy

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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

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.)

Your answer

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) \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) \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 \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 = 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.

Where next?

← Back to the problem: Each band is a perimeter, and its direction says which one