← Rolling matches one circumference to another · Circumference and Area of a Circle

Rolling matches one circumference to another · 12 practice problems

7.G.B.47.G.A.1

Generated variants — 12

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

Variant 1 easy answer: 3 cm

A cone whose slant height is $9\ \text{cm}$ is rolled about its apex, as shown at the right. After $3$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 9 cm rolls about its apex and returns to its start after 3 turns. We want its base's radius.

Givens

The slant height is 9 cm.
It takes 3 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 9 cm from it, so the base runs round a circle of that radius.

9 \times 2 \times 3.14 = 56.52

The path is 56.52 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 3 of them cover it exactly.

56.52 \div 3 = 18.84

The base is 18.84 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

18.84 \div 3.14 \div 2 = 3

The base's radius is 3 cm.

Answer: 3 cm

4 · Reviewdoes it hold up?

The slant height is 3 times the base's radius, 3 cm -- the same 3 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 2 easy answer: 2 cm

A cone whose slant height is $10\ \text{cm}$ is rolled about its apex, as shown at the right. After $5$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 10 cm rolls about its apex and returns to its start after 5 turns. We want its base's radius.

Givens

The slant height is 10 cm.
It takes 5 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 10 cm from it, so the base runs round a circle of that radius.

10 \times 2 \times 3.14 = 62.8

The path is 62.8 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 5 of them cover it exactly.

62.8 \div 5 = 12.56

The base is 12.56 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

12.56 \div 3.14 \div 2 = 2

The base's radius is 2 cm.

Answer: 2 cm

4 · Reviewdoes it hold up?

The slant height is 5 times the base's radius, 2 cm -- the same 5 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 3 easy answer: 3 cm

A cone whose slant height is $12\ \text{cm}$ is rolled about its apex, as shown at the right. After $4$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 12 cm rolls about its apex and returns to its start after 4 turns. We want its base's radius.

Givens

The slant height is 12 cm.
It takes 4 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 12 cm from it, so the base runs round a circle of that radius.

12 \times 2 \times 3.14 = 75.36

The path is 75.36 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 4 of them cover it exactly.

75.36 \div 4 = 18.84

The base is 18.84 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

18.84 \div 3.14 \div 2 = 3

The base's radius is 3 cm.

Answer: 3 cm

4 · Reviewdoes it hold up?

The slant height is 4 times the base's radius, 3 cm -- the same 4 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 4 easy answer: 2 cm

A cone whose slant height is $14\ \text{cm}$ is rolled about its apex, as shown at the right. After $7$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 14 cm rolls about its apex and returns to its start after 7 turns. We want its base's radius.

Givens

The slant height is 14 cm.
It takes 7 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 14 cm from it, so the base runs round a circle of that radius.

14 \times 2 \times 3.14 = 87.92

The path is 87.92 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 7 of them cover it exactly.

87.92 \div 7 = 12.56

The base is 12.56 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

12.56 \div 3.14 \div 2 = 2

The base's radius is 2 cm.

Answer: 2 cm

4 · Reviewdoes it hold up?

The slant height is 7 times the base's radius, 2 cm -- the same 7 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 5 medium answer: 3 cm

A cone whose slant height is $15\ \text{cm}$ is rolled about its apex, as shown at the right. After $5$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 15 cm rolls about its apex and returns to its start after 5 turns. We want its base's radius.

Givens

The slant height is 15 cm.
It takes 5 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 15 cm from it, so the base runs round a circle of that radius.

15 \times 2 \times 3.14 = 94.2

The path is 94.2 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 5 of them cover it exactly.

94.2 \div 5 = 18.84

The base is 18.84 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

18.84 \div 3.14 \div 2 = 3

The base's radius is 3 cm.

Answer: 3 cm

4 · Reviewdoes it hold up?

The slant height is 5 times the base's radius, 3 cm -- the same 5 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 6 medium answer: 2 cm

A cone whose slant height is $16\ \text{cm}$ is rolled about its apex, as shown at the right. After $8$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 16 cm rolls about its apex and returns to its start after 8 turns. We want its base's radius.

Givens

The slant height is 16 cm.
It takes 8 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 16 cm from it, so the base runs round a circle of that radius.

16 \times 2 \times 3.14 = 100.48

The path is 100.48 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 8 of them cover it exactly.

100.48 \div 8 = 12.56

The base is 12.56 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

12.56 \div 3.14 \div 2 = 2

The base's radius is 2 cm.

Answer: 2 cm

4 · Reviewdoes it hold up?

The slant height is 8 times the base's radius, 2 cm -- the same 8 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 7 medium answer: 3 cm

A cone whose slant height is $18\ \text{cm}$ is rolled about its apex, as shown at the right. After $6$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 18 cm rolls about its apex and returns to its start after 6 turns. We want its base's radius.

Givens

The slant height is 18 cm.
It takes 6 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 18 cm from it, so the base runs round a circle of that radius.

18 \times 2 \times 3.14 = 113.04

The path is 113.04 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 6 of them cover it exactly.

113.04 \div 6 = 18.84

The base is 18.84 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

18.84 \div 3.14 \div 2 = 3

The base's radius is 3 cm.

Answer: 3 cm

4 · Reviewdoes it hold up?

The slant height is 6 times the base's radius, 3 cm -- the same 6 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 8 medium answer: 5 cm

A cone whose slant height is $20\ \text{cm}$ is rolled about its apex, as shown at the right. After $4$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 20 cm rolls about its apex and returns to its start after 4 turns. We want its base's radius.

Givens

The slant height is 20 cm.
It takes 4 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 20 cm from it, so the base runs round a circle of that radius.

20 \times 2 \times 3.14 = 125.6

The path is 125.6 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 4 of them cover it exactly.

125.6 \div 4 = 31.4

The base is 31.4 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

31.4 \div 3.14 \div 2 = 5

The base's radius is 5 cm.

Answer: 5 cm

4 · Reviewdoes it hold up?

The slant height is 4 times the base's radius, 5 cm -- the same 4 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 9 hard answer: 8 cm

A cone whose slant height is $24\ \text{cm}$ is rolled about its apex, as shown at the right. After $3$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 24 cm rolls about its apex and returns to its start after 3 turns. We want its base's radius.

Givens

The slant height is 24 cm.
It takes 3 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 24 cm from it, so the base runs round a circle of that radius.

24 \times 2 \times 3.14 = 150.72

The path is 150.72 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 3 of them cover it exactly.

150.72 \div 3 = 50.24

The base is 50.24 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

50.24 \div 3.14 \div 2 = 8

The base's radius is 8 cm.

Answer: 8 cm

4 · Reviewdoes it hold up?

The slant height is 3 times the base's radius, 8 cm -- the same 3 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 10 hard answer: 6 cm

A cone whose slant height is $30\ \text{cm}$ is rolled about its apex, as shown at the right. After $5$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 30 cm rolls about its apex and returns to its start after 5 turns. We want its base's radius.

Givens

The slant height is 30 cm.
It takes 5 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 30 cm from it, so the base runs round a circle of that radius.

30 \times 2 \times 3.14 = 188.4

The path is 188.4 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 5 of them cover it exactly.

188.4 \div 5 = 37.68

The base is 37.68 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

37.68 \div 3.14 \div 2 = 6

The base's radius is 6 cm.

Answer: 6 cm

4 · Reviewdoes it hold up?

The slant height is 5 times the base's radius, 6 cm -- the same 5 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 11 hard answer: 4 cm

A cone whose slant height is $36\ \text{cm}$ is rolled about its apex, as shown at the right. After $9$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 36 cm rolls about its apex and returns to its start after 9 turns. We want its base's radius.

Givens

The slant height is 36 cm.
It takes 9 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 36 cm from it, so the base runs round a circle of that radius.

36 \times 2 \times 3.14 = 226.08

The path is 226.08 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 9 of them cover it exactly.

226.08 \div 9 = 25.12

The base is 25.12 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

25.12 \div 3.14 \div 2 = 4

The base's radius is 4 cm.

Answer: 4 cm

4 · Reviewdoes it hold up?

The slant height is 9 times the base's radius, 4 cm -- the same 9 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.

Variant 12 hard answer: 5 cm

A cone whose slant height is $40\ \text{cm}$ is rolled about its apex, as shown at the right. After $8$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A cone with slant height 40 cm rolls about its apex and returns to its start after 8 turns. We want its base's radius.

Givens

The slant height is 40 cm.
It takes 8 turns to come back round.
The apex stays in one place.

Unknowns

The radius of the cone's base.

Constraints

The cone rolls without slipping.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #11 Work Backwards#7 Identify Subproblems

Rolling puts two circumferences against each other. Find the circle the base travels round -- its radius is the slant height, because the apex never moves -- and share it out among the turns.

3 · Execute3 carry out the plan

1Find the path

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

With the apex fixed, every point of the base stays 40 cm from it, so the base runs round a circle of that radius.

40 \times 2 \times 3.14 = 251.2

The path is 251.2 cm round.

2Share it among the turns

#11 Work Backwards 7.G.A.1

Each turn lays the base's own rim along the path once, and 8 of them cover it exactly.

251.2 \div 8 = 31.4

The base is 31.4 cm round.

3Recover the radius

#7 Identify Subproblems 7.G.B.4

Divide the circumference by 3.14 and then by 2.

31.4 \div 3.14 \div 2 = 5

The base's radius is 5 cm.

Answer: 5 cm

4 · Reviewdoes it hold up?

The slant height is 8 times the base's radius, 5 cm -- the same 8 as the number of turns, which is what matching the two circumferences has to give.

Another way: Comparing the radii straight away works too, and shows why: the number of turns is the slant height divided by the base's radius, with the 3.14 cancelling on both sides.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Moving between a circle's radius and its circumference.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Matching the rolled distance against the path.

💡Takeaway. A rolling shape lays its own edge along the path. Count the turns and the two edges tell you each other's length.