Practice · A perimeter follows the real edge, so a covered side does not count · Sensim Math

Variant 1 easy

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 8 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 8 cm.
The quarter circle is centred at the bottom-left corner, radius 8 cm.
The semicircle stands on the bottom side, radius 4 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 8 cm.

8 \times 2 \times 3.14 \div 4 = 12.56

12.56 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 4 cm -- half the radius but twice the share, so the same length.

4 \times 2 \times 3.14 \div 2 = 12.56

12.56 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

8 + 12.56 + 12.56 = 33.12

33.12 cm round.

Answer: 33.12 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 41.12 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 12.56 cm each, so the perimeter is 8 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 2 easy

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 10 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 10 cm.
The quarter circle is centred at the bottom-left corner, radius 10 cm.
The semicircle stands on the bottom side, radius 5 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 10 cm.

10 \times 2 \times 3.14 \div 4 = 15.7

15.7 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 5 cm -- half the radius but twice the share, so the same length.

5 \times 2 \times 3.14 \div 2 = 15.7

15.7 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

10 + 15.7 + 15.7 = 41.4

41.4 cm round.

Answer: 41.4 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 51.4 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 15.7 cm each, so the perimeter is 10 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 3 easy

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 12 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 12 cm.
The quarter circle is centred at the bottom-left corner, radius 12 cm.
The semicircle stands on the bottom side, radius 6 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 12 cm.

12 \times 2 \times 3.14 \div 4 = 18.84

18.84 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 6 cm -- half the radius but twice the share, so the same length.

6 \times 2 \times 3.14 \div 2 = 18.84

18.84 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

12 + 18.84 + 18.84 = 49.68

49.68 cm round.

Answer: 49.68 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 61.68 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 18.84 cm each, so the perimeter is 12 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 4 easy

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 14 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 14 cm.
The quarter circle is centred at the bottom-left corner, radius 14 cm.
The semicircle stands on the bottom side, radius 7 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 14 cm.

14 \times 2 \times 3.14 \div 4 = 21.98

21.98 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 7 cm -- half the radius but twice the share, so the same length.

7 \times 2 \times 3.14 \div 2 = 21.98

21.98 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

14 + 21.98 + 21.98 = 57.96

57.96 cm round.

Answer: 57.96 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 71.96 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 21.98 cm each, so the perimeter is 14 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 5 medium

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 16 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 16 cm.
The quarter circle is centred at the bottom-left corner, radius 16 cm.
The semicircle stands on the bottom side, radius 8 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 16 cm.

16 \times 2 \times 3.14 \div 4 = 25.12

25.12 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 8 cm -- half the radius but twice the share, so the same length.

8 \times 2 \times 3.14 \div 2 = 25.12

25.12 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

16 + 25.12 + 25.12 = 66.24

66.24 cm round.

Answer: 66.24 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 82.24 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 25.12 cm each, so the perimeter is 16 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 6 medium

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 18 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 18 cm.
The quarter circle is centred at the bottom-left corner, radius 18 cm.
The semicircle stands on the bottom side, radius 9 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 18 cm.

18 \times 2 \times 3.14 \div 4 = 28.26

28.26 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 9 cm -- half the radius but twice the share, so the same length.

9 \times 2 \times 3.14 \div 2 = 28.26

28.26 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

18 + 28.26 + 28.26 = 74.52

74.52 cm round.

Answer: 74.52 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 92.52 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 28.26 cm each, so the perimeter is 18 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 7 medium

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 20 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 20 cm.
The quarter circle is centred at the bottom-left corner, radius 20 cm.
The semicircle stands on the bottom side, radius 10 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 20 cm.

20 \times 2 \times 3.14 \div 4 = 31.4

31.4 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 10 cm -- half the radius but twice the share, so the same length.

10 \times 2 \times 3.14 \div 2 = 31.4

31.4 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

20 + 31.4 + 31.4 = 82.8

82.8 cm round.

Answer: 82.8 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 102.8 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 31.4 cm each, so the perimeter is 20 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 8 medium

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 24 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 24 cm.
The quarter circle is centred at the bottom-left corner, radius 24 cm.
The semicircle stands on the bottom side, radius 12 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 24 cm.

24 \times 2 \times 3.14 \div 4 = 37.68

37.68 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 12 cm -- half the radius but twice the share, so the same length.

12 \times 2 \times 3.14 \div 2 = 37.68

37.68 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

24 + 37.68 + 37.68 = 99.36

99.36 cm round.

Answer: 99.36 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 123.36 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 37.68 cm each, so the perimeter is 24 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 9 hard

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 30 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 30 cm.
The quarter circle is centred at the bottom-left corner, radius 30 cm.
The semicircle stands on the bottom side, radius 15 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 30 cm.

30 \times 2 \times 3.14 \div 4 = 47.1

47.1 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 15 cm -- half the radius but twice the share, so the same length.

15 \times 2 \times 3.14 \div 2 = 47.1

47.1 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

30 + 47.1 + 47.1 = 124.2

124.2 cm round.

Answer: 124.2 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 154.2 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 47.1 cm each, so the perimeter is 30 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 10 hard

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 40 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 40 cm.
The quarter circle is centred at the bottom-left corner, radius 40 cm.
The semicircle stands on the bottom side, radius 20 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 40 cm.

40 \times 2 \times 3.14 \div 4 = 62.8

62.8 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 20 cm -- half the radius but twice the share, so the same length.

20 \times 2 \times 3.14 \div 2 = 62.8

62.8 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

40 + 62.8 + 62.8 = 165.6

165.6 cm round.

Answer: 165.6 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 205.6 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 62.8 cm each, so the perimeter is 40 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 11 hard

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 50 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 50 cm.
The quarter circle is centred at the bottom-left corner, radius 50 cm.
The semicircle stands on the bottom side, radius 25 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 50 cm.

50 \times 2 \times 3.14 \div 4 = 78.5

78.5 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 25 cm -- half the radius but twice the share, so the same length.

25 \times 2 \times 3.14 \div 2 = 78.5

78.5 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

50 + 78.5 + 78.5 = 207

207 cm round.

Answer: 207 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 257 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 78.5 cm each, so the perimeter is 50 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

Variant 12 hard

The figure at the right shows parts of circles drawn inside a square. What is the perimeter of the shaded part, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Inside a 60 cm square, a quarter circle from the bottom-left corner and a semicircle on the bottom side bound a region. We want the distance round it.

Givens

The square's side is 60 cm.
The quarter circle is centred at the bottom-left corner, radius 60 cm.
The semicircle stands on the bottom side, radius 30 cm.

Unknowns

The perimeter of the shaded region.

Constraints

A perimeter follows the actual boundary, not every line drawn.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#16 Count the Complement

Trace the edge once with a finger and count what it actually passes along: one straight stretch and two curves. The square's bottom side is underneath the semicircle, so it is never on the boundary.

3 · Execute4 carry out the plan

1Walk the boundary

#1 Draw a Diagram 7.G.B.4

Up the left side, along the big arc, then back along the semicircle -- and the bottom side is covered, so it is not part of the edge.

\text{left side} + \text{quarter arc} + \text{semicircle}

Three stretches, one of them straight.

2Measure the quarter arc

#7 Identify Subproblems 7.G.B.4

A quarter of a circle of radius 60 cm.

60 \times 2 \times 3.14 \div 4 = 94.2

94.2 cm.

3Measure the semicircle

#7 Identify Subproblems 7.G.B.4

Half a circle of radius 30 cm -- half the radius but twice the share, so the same length.

30 \times 2 \times 3.14 \div 2 = 94.2

94.2 cm, the same again.

4Add the three

#16 Count the Complement 7.G.B.4

The straight side plus the two arcs.

60 + 94.2 + 94.2 = 248.4

248.4 cm round.

Answer: 248.4 cm

4 · Reviewdoes it hold up?

Counting the bottom side as well would give 308.4 cm, but that side is underneath the semicircle and never on the outside of the shaded region.

Another way: The two arcs come to the same 94.2 cm each, so the perimeter is 60 plus twice that -- worth noticing but not worth relying on without checking.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring arcs as shares of a circumference.

💡Takeaway. Trace the edge with your finger. Whatever your finger does not touch is not part of the perimeter.

A perimeter follows the real edge, so a covered side does not count · 12 practice problems

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