← One turn of a wheel covers exactly one circumference · Circumference and Area of a Circle

One turn of a wheel covers exactly one circumference · 12 practice problems

7.G.B.4

Generated variants — 12

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

Variant 1 easy answer: 659.4 cm

A hoop with a radius of $10\ \text{cm}$ is rolled in one direction and makes $10$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 10 cm rolls 10.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 10 cm.
It makes 10.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

10 \times 2 \times 3.14 = 62.8

The rim is 62.8 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 62.8\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

10.5 of those.

62.8 \times 10.5 = 659.4

It travels 659.4 cm.

Answer: 659.4 cm

4 · Reviewdoes it hold up?

659.4 cm is between 596.6 and 722.2 -- between what 9.5 and 11.5 turns would give, as it must be.

Another way: Working from the diameter, 20 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 2 easy answer: 640.56 cm

A hoop with a radius of $12\ \text{cm}$ is rolled in one direction and makes $8$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 12 cm rolls 8.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 12 cm.
It makes 8.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

12 \times 2 \times 3.14 = 75.36

The rim is 75.36 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 75.36\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

8.5 of those.

75.36 \times 8.5 = 640.56

It travels 640.56 cm.

Answer: 640.56 cm

4 · Reviewdoes it hold up?

640.56 cm is between 565.2 and 715.92 -- between what 7.5 and 9.5 turns would give, as it must be.

Another way: Working from the diameter, 24 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 3 easy answer: 612.3 cm

A hoop with a radius of $15\ \text{cm}$ is rolled in one direction and makes $6$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 15 cm rolls 6.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 15 cm.
It makes 6.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

15 \times 2 \times 3.14 = 94.2

The rim is 94.2 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 94.2\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

6.5 of those.

94.2 \times 6.5 = 612.3

It travels 612.3 cm.

Answer: 612.3 cm

4 · Reviewdoes it hold up?

612.3 cm is between 518.1 and 706.5 -- between what 5.5 and 7.5 turns would give, as it must be.

Another way: Working from the diameter, 30 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 4 easy answer: 439.6 cm

A hoop with a radius of $20\ \text{cm}$ is rolled in one direction and makes $3$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 20 cm rolls 3.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 20 cm.
It makes 3.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

20 \times 2 \times 3.14 = 125.6

The rim is 125.6 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 125.6\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

3.5 of those.

125.6 \times 3.5 = 439.6

It travels 439.6 cm.

Answer: 439.6 cm

4 · Reviewdoes it hold up?

439.6 cm is between 314 and 565.2 -- between what 2.5 and 4.5 turns would give, as it must be.

Another way: Working from the diameter, 40 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 5 medium answer: 392.5 cm

A hoop with a radius of $25\ \text{cm}$ is rolled in one direction and makes $2$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 25 cm rolls 2.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 25 cm.
It makes 2.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

25 \times 2 \times 3.14 = 157

The rim is 157 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 157\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

2.5 of those.

157 \times 2.5 = 392.5

It travels 392.5 cm.

Answer: 392.5 cm

4 · Reviewdoes it hold up?

392.5 cm is between 235.5 and 549.5 -- between what 1.5 and 3.5 turns would give, as it must be.

Another way: Working from the diameter, 50 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 6 medium answer: 791.28 cm

A hoop with a radius of $28\ \text{cm}$ is rolled in one direction and makes $4$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 28 cm rolls 4.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 28 cm.
It makes 4.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

28 \times 2 \times 3.14 = 175.84

The rim is 175.84 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 175.84\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

4.5 of those.

175.84 \times 4.5 = 791.28

It travels 791.28 cm.

Answer: 791.28 cm

4 · Reviewdoes it hold up?

791.28 cm is between 615.44 and 967.12 -- between what 3.5 and 5.5 turns would give, as it must be.

Another way: Working from the diameter, 56 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 7 medium answer: 847.8 cm

A hoop with a radius of $30\ \text{cm}$ is rolled in one direction and makes $4$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 30 cm rolls 4.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 30 cm.
It makes 4.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

30 \times 2 \times 3.14 = 188.4

The rim is 188.4 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 188.4\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

4.5 of those.

188.4 \times 4.5 = 847.8

It travels 847.8 cm.

Answer: 847.8 cm

4 · Reviewdoes it hold up?

847.8 cm is between 659.4 and 1036.2 -- between what 3.5 and 5.5 turns would give, as it must be.

Another way: Working from the diameter, 60 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 8 medium answer: 1648.5 cm

A hoop with a radius of $35\ \text{cm}$ is rolled in one direction and makes $7$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 35 cm rolls 7.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 35 cm.
It makes 7.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

35 \times 2 \times 3.14 = 219.8

The rim is 219.8 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 219.8\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

7.5 of those.

219.8 \times 7.5 = 1648.5

It travels 1648.5 cm.

Answer: 1648.5 cm

4 · Reviewdoes it hold up?

1648.5 cm is between 1428.7 and 1868.3 -- between what 6.5 and 8.5 turns would give, as it must be.

Another way: Working from the diameter, 70 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 9 hard answer: 1381.6 cm

A hoop with a radius of $40\ \text{cm}$ is rolled in one direction and makes $5$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 40 cm rolls 5.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 40 cm.
It makes 5.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

40 \times 2 \times 3.14 = 251.2

The rim is 251.2 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 251.2\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

5.5 of those.

251.2 \times 5.5 = 1381.6

It travels 1381.6 cm.

Answer: 1381.6 cm

4 · Reviewdoes it hold up?

1381.6 cm is between 1130.4 and 1632.8 -- between what 4.5 and 6.5 turns would give, as it must be.

Another way: Working from the diameter, 80 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 10 hard answer: 989.1 cm

A hoop with a radius of $45\ \text{cm}$ is rolled in one direction and makes $3$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 45 cm rolls 3.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 45 cm.
It makes 3.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

45 \times 2 \times 3.14 = 282.6

The rim is 282.6 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 282.6\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

3.5 of those.

282.6 \times 3.5 = 989.1

It travels 989.1 cm.

Answer: 989.1 cm

4 · Reviewdoes it hold up?

989.1 cm is between 706.5 and 1271.7 -- between what 2.5 and 4.5 turns would give, as it must be.

Another way: Working from the diameter, 90 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 11 hard answer: 471 cm

A hoop with a radius of $50\ \text{cm}$ is rolled in one direction and makes $1$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 50 cm rolls 1.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 50 cm.
It makes 1.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

50 \times 2 \times 3.14 = 314

The rim is 314 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 314\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

1.5 of those.

314 \times 1.5 = 471

It travels 471 cm.

Answer: 471 cm

4 · Reviewdoes it hold up?

471 cm is between 157 and 785 -- between what 0.5 and 2.5 turns would give, as it must be.

Another way: Working from the diameter, 100 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.

Variant 12 hard answer: 942 cm

A hoop with a radius of $60\ \text{cm}$ is rolled in one direction and makes $2$ and a half turns. How far does it travel, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A hoop of radius 60 cm rolls 2.5 turns without slipping. We want the distance it covers.

Givens

The hoop's radius is 60 cm.
It makes 2.5 turns.

Unknowns

The distance the hoop travels.

Constraints

It rolls without slipping, so the rim lays itself out along the ground.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#8 Analyze the Units

Picture one turn: every point of the rim touches the ground exactly once, so the hoop moves forward by its own circumference. After that the distance is one multiplication.

3 · Execute3 carry out the plan

1Find the circumference

#7 Identify Subproblems 7.G.B.4

The radius is given, so double it for the diameter and multiply by 3.14.

60 \times 2 \times 3.14 = 376.8

The rim is 376.8 cm long.

2See what one turn covers

#17 Visualize Spatial Relationships 7.G.B.4

Rolling without slipping lays the whole rim along the ground once per turn.

1 \text{ turn} = 376.8\ \text{cm}

One turn, one circumference.

3Multiply by the turns

#8 Analyze the Units 7.G.B.4

2.5 of those.

376.8 \times 2.5 = 942

It travels 942 cm.

Answer: 942 cm

4 · Reviewdoes it hold up?

942 cm is between 565.2 and 1318.8 -- between what 1.5 and 3.5 turns would give, as it must be.

Another way: Working from the diameter, 120 cm, straight to the circumference skips one step but hides why the doubling was needed.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding the circumference and using it as the distance per turn.

💡Takeaway. A wheel that rolls without slipping moves forward by its own distance round, once for every turn.