Practice · Measure an awkward leftover by subtracting what you can measure · Sensim Math

Variant 1 easy

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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

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.)

Your answer

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.

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

Generated variants — 12

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?