← A circle grows in two directions, so its area goes with the square · Circumference and Area of a Circle

A circle grows in two directions, so its area goes with the square · 12 practice problems

7.G.B.47.RP.A.27.G.A.1

Generated variants — 12

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

Variant 1 easy answer: 100 g

Making a round pizza with a diameter of $10\ \text{cm}$ takes $25\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $20\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 10 cm pizza takes 25 g of flour. We want the flour for a 20 cm pizza of the same thickness.

Givens

A pizza 10 cm across takes 25 g of flour.
The other pizza is 20 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 20 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 5 cm and 10 cm.

78.5,\quad 314

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 2 times bigger, and the area 4 times -- that factor squared.

314 \div 78.5 = 4

4 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

25 \times 4 = 100

100 g of flour.

Answer: 100 g

4 · Reviewdoes it hold up?

The pizza is only 2 times wider but 4 times heavier, and 2 squared is 4 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 50 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 2 easy answer: 45 g

Making a round pizza with a diameter of $12\ \text{cm}$ takes $5\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $36\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 12 cm pizza takes 5 g of flour. We want the flour for a 36 cm pizza of the same thickness.

Givens

A pizza 12 cm across takes 5 g of flour.
The other pizza is 36 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 36 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 6 cm and 18 cm.

113.04,\quad 1017.36

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 3 times bigger, and the area 9 times -- that factor squared.

1017.36 \div 113.04 = 9

9 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

5 \times 9 = 45

45 g of flour.

Answer: 45 g

4 · Reviewdoes it hold up?

The pizza is only 3 times wider but 9 times heavier, and 3 squared is 9 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 15 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 3 easy answer: 100 g

Making a round pizza with a diameter of $18\ \text{cm}$ takes $25\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $36\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 18 cm pizza takes 25 g of flour. We want the flour for a 36 cm pizza of the same thickness.

Givens

A pizza 18 cm across takes 25 g of flour.
The other pizza is 36 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 36 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 9 cm and 18 cm.

254.34,\quad 1017.36

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 2 times bigger, and the area 4 times -- that factor squared.

1017.36 \div 254.34 = 4

4 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

25 \times 4 = 100

100 g of flour.

Answer: 100 g

4 · Reviewdoes it hold up?

The pizza is only 2 times wider but 4 times heavier, and 2 squared is 4 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 50 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 4 easy answer: 405 g

Making a round pizza with a diameter of $8\ \text{cm}$ takes $45\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $24\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 8 cm pizza takes 45 g of flour. We want the flour for a 24 cm pizza of the same thickness.

Givens

A pizza 8 cm across takes 45 g of flour.
The other pizza is 24 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 24 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 4 cm and 12 cm.

50.24,\quad 452.16

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 3 times bigger, and the area 9 times -- that factor squared.

452.16 \div 50.24 = 9

9 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

45 \times 9 = 405

405 g of flour.

Answer: 405 g

4 · Reviewdoes it hold up?

The pizza is only 3 times wider but 9 times heavier, and 3 squared is 9 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 135 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 5 medium answer: 540 g

Making a round pizza with a diameter of $10\ \text{cm}$ takes $60\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $30\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 10 cm pizza takes 60 g of flour. We want the flour for a 30 cm pizza of the same thickness.

Givens

A pizza 10 cm across takes 60 g of flour.
The other pizza is 30 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 30 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 5 cm and 15 cm.

78.5,\quad 706.5

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 3 times bigger, and the area 9 times -- that factor squared.

706.5 \div 78.5 = 9

9 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

60 \times 9 = 540

540 g of flour.

Answer: 540 g

4 · Reviewdoes it hold up?

The pizza is only 3 times wider but 9 times heavier, and 3 squared is 9 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 180 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 6 medium answer: 315 g

Making a round pizza with a diameter of $20\ \text{cm}$ takes $35\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $60\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 20 cm pizza takes 35 g of flour. We want the flour for a 60 cm pizza of the same thickness.

Givens

A pizza 20 cm across takes 35 g of flour.
The other pizza is 60 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 60 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 10 cm and 30 cm.

314,\quad 2826

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 3 times bigger, and the area 9 times -- that factor squared.

2826 \div 314 = 9

9 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

35 \times 9 = 315

315 g of flour.

Answer: 315 g

4 · Reviewdoes it hold up?

The pizza is only 3 times wider but 9 times heavier, and 3 squared is 9 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 105 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 7 medium answer: 960 g

Making a round pizza with a diameter of $8\ \text{cm}$ takes $60\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $32\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 8 cm pizza takes 60 g of flour. We want the flour for a 32 cm pizza of the same thickness.

Givens

A pizza 8 cm across takes 60 g of flour.
The other pizza is 32 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 32 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 4 cm and 16 cm.

50.24,\quad 803.84

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 4 times bigger, and the area 16 times -- that factor squared.

803.84 \div 50.24 = 16

16 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

60 \times 16 = 960

960 g of flour.

Answer: 960 g

4 · Reviewdoes it hold up?

The pizza is only 4 times wider but 16 times heavier, and 4 squared is 16 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 240 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 8 hard answer: 720 g

Making a round pizza with a diameter of $18\ \text{cm}$ takes $80\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $54\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 18 cm pizza takes 80 g of flour. We want the flour for a 54 cm pizza of the same thickness.

Givens

A pizza 18 cm across takes 80 g of flour.
The other pizza is 54 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 54 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 9 cm and 27 cm.

254.34,\quad 2289.06

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 3 times bigger, and the area 9 times -- that factor squared.

2289.06 \div 254.34 = 9

9 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

80 \times 9 = 720

720 g of flour.

Answer: 720 g

4 · Reviewdoes it hold up?

The pizza is only 3 times wider but 9 times heavier, and 3 squared is 9 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 240 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 9 medium answer: 320 g

Making a round pizza with a diameter of $20\ \text{cm}$ takes $20\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $80\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 20 cm pizza takes 20 g of flour. We want the flour for a 80 cm pizza of the same thickness.

Givens

A pizza 20 cm across takes 20 g of flour.
The other pizza is 80 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 80 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 10 cm and 40 cm.

314,\quad 5024

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 4 times bigger, and the area 16 times -- that factor squared.

5024 \div 314 = 16

16 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

20 \times 16 = 320

320 g of flour.

Answer: 320 g

4 · Reviewdoes it hold up?

The pizza is only 4 times wider but 16 times heavier, and 4 squared is 16 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 80 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 10 hard answer: 1200 g

Making a round pizza with a diameter of $20\ \text{cm}$ takes $75\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $80\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 20 cm pizza takes 75 g of flour. We want the flour for a 80 cm pizza of the same thickness.

Givens

A pizza 20 cm across takes 75 g of flour.
The other pizza is 80 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 80 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 10 cm and 40 cm.

314,\quad 5024

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 4 times bigger, and the area 16 times -- that factor squared.

5024 \div 314 = 16

16 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

75 \times 16 = 1200

1200 g of flour.

Answer: 1200 g

4 · Reviewdoes it hold up?

The pizza is only 4 times wider but 16 times heavier, and 4 squared is 16 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 300 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 11 hard answer: 765 g

Making a round pizza with a diameter of $4\ \text{cm}$ takes $85\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $12\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 4 cm pizza takes 85 g of flour. We want the flour for a 12 cm pizza of the same thickness.

Givens

A pizza 4 cm across takes 85 g of flour.
The other pizza is 12 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 12 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 2 cm and 6 cm.

12.56,\quad 113.04

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 3 times bigger, and the area 9 times -- that factor squared.

113.04 \div 12.56 = 9

9 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

85 \times 9 = 765

765 g of flour.

Answer: 765 g

4 · Reviewdoes it hold up?

The pizza is only 3 times wider but 9 times heavier, and 3 squared is 9 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 255 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.

Variant 12 hard answer: 360 g

Making a round pizza with a diameter of $16\ \text{cm}$ takes $90\ \text{g}$ of flour. How many grams of flour are needed to make a pizza of the same thickness with a diameter of $32\ \text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A 16 cm pizza takes 90 g of flour. We want the flour for a 32 cm pizza of the same thickness.

Givens

A pizza 16 cm across takes 90 g of flour.
The other pizza is 32 cm across.
Both are the same thickness.

Unknowns

The flour needed for the 32 cm pizza.

Constraints

Same thickness, so the flour is proportional to the area.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Equal thickness means the flour follows the area, not the width. So compare the two areas rather than the two diameters -- a circle grows in two directions at once.

3 · Execute4 carry out the plan

1See what the flour follows

#8 Analyze the Units 7.RP.A.2

Same thickness, so the amount of dough is the area of the base times a fixed depth.

\text{flour} \propto \text{area}

Area, not diameter.

2Find both areas

#7 Identify Subproblems 7.G.B.4

Radii 8 cm and 16 cm.

200.96,\quad 803.84

Two areas to compare.

3Compare them

#9 Solve an Easier Related Problem 7.G.A.1

The diameter is 2 times bigger, and the area 4 times -- that factor squared.

803.84 \div 200.96 = 4

4 times the area.

4Scale the flour

#8 Analyze the Units 7.RP.A.2

The flour grows by the same factor as the area.

90 \times 4 = 360

360 g of flour.

Answer: 360 g

4 · Reviewdoes it hold up?

The pizza is only 2 times wider but 4 times heavier, and 2 squared is 4 -- which is what growing in two directions means.

Another way: Scaling the flour by the diameter ratio instead would give 180 g, far too little: it treats the pizza as if it only got wider and not longer as well.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Finding each circle's area from its diameter.
7.RP.A.2 Recognize and represent proportional relationships between quantities — Treating the flour as proportional to the area.
7.G.A.1 Solve problems involving scale drawings of geometric figures — Relating the scale factor of the lengths to that of the areas.

💡Takeaway. Twice as wide is four times as big. A circle grows in two directions, and the flour follows the area.