← Measure an awkward leftover by subtracting what you can measure · Circumference and Area of a Circle

Measure an awkward leftover by subtracting what you can measure · 12 practice problems

7.G.B.46.G.A.1

Generated variants — 12

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

Variant 1 easy answer: 13.76 cm²

The figure at the right is the top view of 4 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $12.56\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

4 equal circles are packed 2 by 2 inside a square. Each is 12.56 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 12.56 cm.
The circles are packed 2 by 2, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

12.56 \div 3.14 = 4,\quad \text{radius} = 2

Each circle is 4 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

2 circles sit side by side along it, each touching the next and the wall.

4 \times 2 = 8

The square is 8 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 4 circles.

8 \times 8 = 64,\quad 12.56 \times 4 = 50.24

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

64 - 50.24 = 13.76

13.76 cm2 is empty.

Answer: 13.76 cm²

4 · Reviewdoes it hold up?

The circles cover 50.24 of the square's 64 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 4 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 2 easy answer: 30.96 cm²

The figure at the right is the top view of 9 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $12.56\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

9 equal circles are packed 3 by 3 inside a square. Each is 12.56 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 12.56 cm.
The circles are packed 3 by 3, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

12.56 \div 3.14 = 4,\quad \text{radius} = 2

Each circle is 4 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

3 circles sit side by side along it, each touching the next and the wall.

4 \times 3 = 12

The square is 12 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 9 circles.

12 \times 12 = 144,\quad 12.56 \times 9 = 113.04

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

144 - 113.04 = 30.96

30.96 cm2 is empty.

Answer: 30.96 cm²

4 · Reviewdoes it hold up?

The circles cover 113.04 of the square's 144 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 9 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 3 easy answer: 55.04 cm²

The figure at the right is the top view of 16 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $12.56\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

16 equal circles are packed 4 by 4 inside a square. Each is 12.56 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 12.56 cm.
The circles are packed 4 by 4, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

12.56 \div 3.14 = 4,\quad \text{radius} = 2

Each circle is 4 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

4 circles sit side by side along it, each touching the next and the wall.

4 \times 4 = 16

The square is 16 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 16 circles.

16 \times 16 = 256,\quad 12.56 \times 16 = 200.96

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

256 - 200.96 = 55.04

55.04 cm2 is empty.

Answer: 55.04 cm²

4 · Reviewdoes it hold up?

The circles cover 200.96 of the square's 256 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 16 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 4 easy answer: 30.96 cm²

The figure at the right is the top view of 4 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $18.84\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

4 equal circles are packed 2 by 2 inside a square. Each is 18.84 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 18.84 cm.
The circles are packed 2 by 2, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

18.84 \div 3.14 = 6,\quad \text{radius} = 3

Each circle is 6 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

2 circles sit side by side along it, each touching the next and the wall.

6 \times 2 = 12

The square is 12 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 4 circles.

12 \times 12 = 144,\quad 28.26 \times 4 = 113.04

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

144 - 113.04 = 30.96

30.96 cm2 is empty.

Answer: 30.96 cm²

4 · Reviewdoes it hold up?

The circles cover 113.04 of the square's 144 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 4 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 5 medium answer: 278.64 cm²

The figure at the right is the top view of 9 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $37.68\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

9 equal circles are packed 3 by 3 inside a square. Each is 37.68 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 37.68 cm.
The circles are packed 3 by 3, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

37.68 \div 3.14 = 12,\quad \text{radius} = 6

Each circle is 12 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

3 circles sit side by side along it, each touching the next and the wall.

12 \times 3 = 36

The square is 36 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 9 circles.

36 \times 36 = 1296,\quad 113.04 \times 9 = 1017.36

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

1296 - 1017.36 = 278.64

278.64 cm2 is empty.

Answer: 278.64 cm²

4 · Reviewdoes it hold up?

The circles cover 1017.36 of the square's 1296 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 9 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 6 medium answer: 168.56 cm²

The figure at the right is the top view of 4 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $43.96\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

4 equal circles are packed 2 by 2 inside a square. Each is 43.96 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 43.96 cm.
The circles are packed 2 by 2, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

43.96 \div 3.14 = 14,\quad \text{radius} = 7

Each circle is 14 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

2 circles sit side by side along it, each touching the next and the wall.

14 \times 2 = 28

The square is 28 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 4 circles.

28 \times 28 = 784,\quad 153.86 \times 4 = 615.44

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

784 - 615.44 = 168.56

168.56 cm2 is empty.

Answer: 168.56 cm²

4 · Reviewdoes it hold up?

The circles cover 615.44 of the square's 784 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 4 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 7 medium answer: 278.64 cm²

The figure at the right is the top view of 4 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $56.52\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

4 equal circles are packed 2 by 2 inside a square. Each is 56.52 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 56.52 cm.
The circles are packed 2 by 2, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

56.52 \div 3.14 = 18,\quad \text{radius} = 9

Each circle is 18 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

2 circles sit side by side along it, each touching the next and the wall.

18 \times 2 = 36

The square is 36 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 4 circles.

36 \times 36 = 1296,\quad 254.34 \times 4 = 1017.36

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

1296 - 1017.36 = 278.64

278.64 cm2 is empty.

Answer: 278.64 cm²

4 · Reviewdoes it hold up?

The circles cover 1017.36 of the square's 1296 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 4 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 8 medium answer: 1376 cm²

The figure at the right is the top view of 16 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $62.8\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

16 equal circles are packed 4 by 4 inside a square. Each is 62.8 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 62.8 cm.
The circles are packed 4 by 4, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

62.8 \div 3.14 = 20,\quad \text{radius} = 10

Each circle is 20 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

4 circles sit side by side along it, each touching the next and the wall.

20 \times 4 = 80

The square is 80 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 16 circles.

80 \times 80 = 6400,\quad 314 \times 16 = 5024

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

6400 - 5024 = 1376

1376 cm2 is empty.

Answer: 1376 cm²

4 · Reviewdoes it hold up?

The circles cover 5024 of the square's 6400 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 16 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 9 hard answer: 936.54 cm²

The figure at the right is the top view of 9 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $69.08\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

9 equal circles are packed 3 by 3 inside a square. Each is 69.08 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 69.08 cm.
The circles are packed 3 by 3, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

69.08 \div 3.14 = 22,\quad \text{radius} = 11

Each circle is 22 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

3 circles sit side by side along it, each touching the next and the wall.

22 \times 3 = 66

The square is 66 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 9 circles.

66 \times 66 = 4356,\quad 379.94 \times 9 = 3419.46

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

4356 - 3419.46 = 936.54

936.54 cm2 is empty.

Answer: 936.54 cm²

4 · Reviewdoes it hold up?

The circles cover 3419.46 of the square's 4356 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 9 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 10 hard answer: 1114.56 cm²

The figure at the right is the top view of 9 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $75.36\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

9 equal circles are packed 3 by 3 inside a square. Each is 75.36 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 75.36 cm.
The circles are packed 3 by 3, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

75.36 \div 3.14 = 24,\quad \text{radius} = 12

Each circle is 24 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

3 circles sit side by side along it, each touching the next and the wall.

24 \times 3 = 72

The square is 72 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 9 circles.

72 \times 72 = 5184,\quad 452.16 \times 9 = 4069.44

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

5184 - 4069.44 = 1114.56

1114.56 cm2 is empty.

Answer: 1114.56 cm²

4 · Reviewdoes it hold up?

The circles cover 4069.44 of the square's 5184 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 9 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 11 hard answer: 1517.04 cm²

The figure at the right is the top view of 9 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $87.92\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

9 equal circles are packed 3 by 3 inside a square. Each is 87.92 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 87.92 cm.
The circles are packed 3 by 3, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

87.92 \div 3.14 = 28,\quad \text{radius} = 14

Each circle is 28 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

3 circles sit side by side along it, each touching the next and the wall.

28 \times 3 = 84

The square is 84 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 9 circles.

84 \times 84 = 7056,\quad 615.44 \times 9 = 5538.96

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

7056 - 5538.96 = 1517.04

1517.04 cm2 is empty.

Answer: 1517.04 cm²

4 · Reviewdoes it hold up?

The circles cover 5538.96 of the square's 7056 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 9 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.

Variant 12 hard answer: 774 cm²

The figure at the right is the top view of 4 identical cylinders packed snugly into a square box; each cylinder's base has a circumference of $94.2\ \text{cm}$ . What is the area of the empty space, in $\text{cm}^2$ ? (Ignore the thickness of the box.)

Show solution

1 · Understandwhat's really being asked

4 equal circles are packed 2 by 2 inside a square. Each is 94.2 cm round, and we want the area the circles leave empty.

Givens

Each circle's circumference is 94.2 cm.
The circles are packed 2 by 2, touching each other and the square.

Unknowns

The area of the empty space inside the square.

Constraints

The circles touch the square's sides, so the packing fixes the square's size.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #11 Work Backwards#7 Identify Subproblems

The empty space is a shape with no formula, so do not measure it directly. Measure the square and the circles, which both have formulas, and subtract.

3 · Execute4 carry out the plan

1Recover the circle from its circumference

#11 Work Backwards 7.G.B.4

Dividing the way round by 3.14 gives the diameter.

94.2 \div 3.14 = 30,\quad \text{radius} = 15

Each circle is 30 cm across.

2Get the square's side

#7 Identify Subproblems 6.G.A.1

2 circles sit side by side along it, each touching the next and the wall.

30 \times 2 = 60

The square is 60 cm on a side.

3Measure both shapes

#7 Identify Subproblems 7.G.B.4

The square, and then all 4 circles.

60 \times 60 = 3600,\quad 706.5 \times 4 = 2826

Two areas, both easy.

4Subtract

#16 Count the Complement 6.G.A.1

What is left of the square is the empty space.

3600 - 2826 = 774

774 cm2 is empty.

Answer: 774 cm²

4 · Reviewdoes it hold up?

The circles cover 2826 of the square's 3600 cm2, about 78 percent -- which is what circles packed in a square always come to, whatever their size.

Another way: Working with one circle and its surrounding square first and multiplying by 4 gives the same answer, since the packing repeats.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Going from circumference to diameter and then to area.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Measuring the square and subtracting the circles.

💡Takeaway. When the shape you want has no formula, measure the shapes around it and take them away.