Practice · Perimeter of the nth figure by rule · Sensim Math

Variant 1 medium

Unit squares with a side length of $1\,\text{cm}$ are joined together without gaps or overlaps following the pattern shown below. What is the perimeter of the ninth figure, in $\text{cm}$ ?

Figure description: Small squares, each with a side length of $1\,\text{cm}$ , are joined edge to edge to build larger and larger square shapes. The first figure is a single small square ( $1\times1$ ), the second figure is a $2\times2$ square made of $4$ small squares, and the third figure is a $3\times3$ square made of $9$ small squares; the pattern continues in the same way ( $\cdots$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Unit squares of side 1 cm are joined to build bigger and bigger squares: figure 1 is a 1x1 square, figure 2 a 2x2 square, figure 3 a 3x3 square, and so on. I must find the perimeter of the ninth figure.

Givens

Each unit square has a side length of 1 cm.
Figure 1 is a 1x1 square (1 unit square).
Figure 2 is a 2x2 square (4 unit squares).
Figure 3 is a 3x3 square (9 unit squares).
The pattern continues the same way, so figure n is an n-by-n square.

Unknowns

The perimeter of the ninth figure, in cm.

Constraints

Figure n is a square whose side is n unit squares, i.e. n cm long.
Perimeter means the total length around the outside.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Each figure is a square, so I first solve the easier related problem of finding each figure's side length, spot the pattern that figure n has side n cm, and then apply the square-perimeter rule (4 times the side) to figure 9.

3 · Execute3 carry out the plan

1Find the side-length pattern

#5 Look for a Pattern 4.OA.C.5

Figure 1 has side 1 cm, figure 2 has side 2 cm, figure 3 has side 3 cm. The side of figure n equals n cm, so the ninth figure has side 9 cm.

\text{side of figure } n = n \text{ cm} \Rightarrow \text{figure 9 side} = 9 \text{ cm}

Spotting that the side grows by 1 cm each step is the Grade 4 idea of generating a shape pattern from a rule.

2Use the perimeter of each small case

#9 Solve an Easier Related Problem 3.MD.D.8

Check the rule on easy cases: figure 1 perimeter = 4 x 1 = 4 cm, figure 2 = 4 x 2 = 8 cm, figure 3 = 4 x 3 = 12 cm. The perimeter is always 4 times the side.

4 \times 1 = 4, \quad 4 \times 2 = 8, \quad 4 \times 3 = 12

Working the small figures first shows the simple 'four equal sides' perimeter rule before jumping to the target figure.

3Compute figure 9's perimeter

#1 Draw a Diagram 4.MD.A.3

The ninth figure is a 9 cm by 9 cm square, so its perimeter is 4 times 9 cm.

4 \times 9 = 36

Multiplying side by 4 for a square's perimeter is a basic Grade 4 rectangle-perimeter formula.

Answer: 36 cm

4 · Reviewdoes it hold up?

The perimeters grow 4, 8, 12, ..., increasing by 4 cm each step, which fits a side that grows by 1 cm each step; 36 cm for figure 9 continues this pattern exactly (4 x 9 = 36). Only the outer edge counts, so the inner grid lines correctly do not add to the perimeter.

Another way: Evaluate finite differences (tool 14): the perimeters 4, 8, 12, ... rise by a constant 4 each time, so the ninth term is 4 x 9 = 36 cm, confirming the result.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing that figure n is an n-by-n square and extending the pattern to figure 9.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding each square figure's perimeter from its side length.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Applying perimeter = 4 x side to the 9 cm square.

💡Takeaway. Figure number tells you the side in cm, and a square's perimeter is just 4 times its side - so figure 9 is 4 x 9 = 36 cm!

Variant 2 medium

Unit squares with a side length of $1\,\text{cm}$ are joined together without gaps or overlaps following the pattern shown below. What is the perimeter of the twelfth figure, in $\text{cm}$ ?

Figure description: Small squares, each with a side length of $1\,\text{cm}$ , are joined edge to edge to build larger and larger square shapes. The first figure is a single small square ( $1\times1$ ), the second figure is a $2\times2$ square made of $4$ small squares, and the third figure is a $3\times3$ square made of $9$ small squares; the pattern continues in the same way ( $\cdots$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Unit squares of side 1 cm are joined to build bigger and bigger squares: figure 1 is a 1x1 square, figure 2 a 2x2 square, figure 3 a 3x3 square, and so on. I must find the perimeter of the twelfth figure.

Givens

Each unit square has a side length of 1 cm.
Figure 1 is a 1x1 square (1 unit square).
Figure 2 is a 2x2 square (4 unit squares).
Figure 3 is a 3x3 square (9 unit squares).
The pattern continues the same way, so figure n is an n-by-n square.

Unknowns

The perimeter of the twelfth figure, in cm.

Constraints

Figure n is a square whose side is n unit squares, i.e. n cm long.
Perimeter means the total length around the outside.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Each figure is a square, so I first solve the easier related problem of finding each figure's side length, spot the pattern that figure n has side n cm, and then apply the square-perimeter rule (4 times the side) to figure 12.

3 · Execute3 carry out the plan

1Find the side-length pattern

#5 Look for a Pattern 4.OA.C.5

Figure 1 has side 1 cm, figure 2 has side 2 cm, figure 3 has side 3 cm. The side of figure n equals n cm, so the twelfth figure has side 12 cm.

\text{side of figure } n = n \text{ cm} \Rightarrow \text{figure 12 side} = 12 \text{ cm}

Spotting that the side grows by 1 cm each step is the Grade 4 idea of generating a shape pattern from a rule.

2Use the perimeter of each small case

#9 Solve an Easier Related Problem 3.MD.D.8

Check the rule on easy cases: figure 1 perimeter = 4 x 1 = 4 cm, figure 2 = 4 x 2 = 8 cm, figure 3 = 4 x 3 = 12 cm. The perimeter is always 4 times the side.

4 \times 1 = 4, \quad 4 \times 2 = 8, \quad 4 \times 3 = 12

Working the small figures first shows the simple 'four equal sides' perimeter rule before jumping to the target figure.

3Compute figure 12's perimeter

#1 Draw a Diagram 4.MD.A.3

The twelfth figure is a 12 cm by 12 cm square, so its perimeter is 4 times 12 cm.

4 \times 12 = 48

Multiplying side by 4 for a square's perimeter is a basic Grade 4 rectangle-perimeter formula.

Answer: 48 cm

4 · Reviewdoes it hold up?

The perimeters grow 4, 8, 12, ..., increasing by 4 cm each step, which fits a side that grows by 1 cm each step; 48 cm for figure 12 continues this pattern exactly (4 x 12 = 48). Only the outer edge counts, so the inner grid lines correctly do not add to the perimeter.

Another way: Evaluate finite differences (tool 14): the perimeters 4, 8, 12, ... rise by a constant 4 each time, so the twelfth term is 4 x 12 = 48 cm, confirming the result.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing that figure n is an n-by-n square and extending the pattern to figure 12.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding each square figure's perimeter from its side length.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Applying perimeter = 4 x side to the 12 cm square.

💡Takeaway. Figure number tells you the side in cm, and a square's perimeter is just 4 times its side - so figure 12 is 4 x 12 = 48 cm!

Variant 3 medium

Unit squares with a side length of $1\,\text{cm}$ are joined together without gaps or overlaps following the pattern shown below. What is the perimeter of the sixth figure, in $\text{cm}$ ?

Figure description: Small squares, each with a side length of $1\,\text{cm}$ , are joined edge to edge to build larger and larger square shapes. The first figure is a single small square ( $1\times1$ ), the second figure is a $2\times2$ square made of $4$ small squares, and the third figure is a $3\times3$ square made of $9$ small squares; the pattern continues in the same way ( $\cdots$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Unit squares of side 1 cm are joined to build bigger and bigger squares: figure 1 is a 1x1 square, figure 2 a 2x2 square, figure 3 a 3x3 square, and so on. I must find the perimeter of the sixth figure.

Givens

Each unit square has a side length of 1 cm.
Figure 1 is a 1x1 square (1 unit square).
Figure 2 is a 2x2 square (4 unit squares).
Figure 3 is a 3x3 square (9 unit squares).
The pattern continues the same way, so figure n is an n-by-n square.

Unknowns

The perimeter of the sixth figure, in cm.

Constraints

Figure n is a square whose side is n unit squares, i.e. n cm long.
Perimeter means the total length around the outside.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Each figure is a square, so I first solve the easier related problem of finding each figure's side length, spot the pattern that figure n has side n cm, and then apply the square-perimeter rule (4 times the side) to figure 6.

3 · Execute3 carry out the plan

1Find the side-length pattern

#5 Look for a Pattern 4.OA.C.5

Figure 1 has side 1 cm, figure 2 has side 2 cm, figure 3 has side 3 cm. The side of figure n equals n cm, so the sixth figure has side 6 cm.

\text{side of figure } n = n \text{ cm} \Rightarrow \text{figure 6 side} = 6 \text{ cm}

Spotting that the side grows by 1 cm each step is the Grade 4 idea of generating a shape pattern from a rule.

2Use the perimeter of each small case

#9 Solve an Easier Related Problem 3.MD.D.8

Check the rule on easy cases: figure 1 perimeter = 4 x 1 = 4 cm, figure 2 = 4 x 2 = 8 cm, figure 3 = 4 x 3 = 12 cm. The perimeter is always 4 times the side.

4 \times 1 = 4, \quad 4 \times 2 = 8, \quad 4 \times 3 = 12

Working the small figures first shows the simple 'four equal sides' perimeter rule before jumping to the target figure.

3Compute figure 6's perimeter

#1 Draw a Diagram 4.MD.A.3

The sixth figure is a 6 cm by 6 cm square, so its perimeter is 4 times 6 cm.

4 \times 6 = 24

Multiplying side by 4 for a square's perimeter is a basic Grade 4 rectangle-perimeter formula.

Answer: 24 cm

4 · Reviewdoes it hold up?

The perimeters grow 4, 8, 12, ..., increasing by 4 cm each step, which fits a side that grows by 1 cm each step; 24 cm for figure 6 continues this pattern exactly (4 x 6 = 24). Only the outer edge counts, so the inner grid lines correctly do not add to the perimeter.

Another way: Evaluate finite differences (tool 14): the perimeters 4, 8, 12, ... rise by a constant 4 each time, so the sixth term is 4 x 6 = 24 cm, confirming the result.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing that figure n is an n-by-n square and extending the pattern to figure 6.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding each square figure's perimeter from its side length.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Applying perimeter = 4 x side to the 6 cm square.

💡Takeaway. Figure number tells you the side in cm, and a square's perimeter is just 4 times its side - so figure 6 is 4 x 6 = 24 cm!

Variant 4 medium

Unit squares with a side length of $1\,\text{cm}$ are joined together without gaps or overlaps following the pattern shown below. What is the perimeter of the eighth figure, in $\text{cm}$ ?

Figure description: Small squares, each with a side length of $1\,\text{cm}$ , are joined edge to edge to build larger and larger square shapes. The first figure is a single small square ( $1\times1$ ), the second figure is a $2\times2$ square made of $4$ small squares, and the third figure is a $3\times3$ square made of $9$ small squares; the pattern continues in the same way ( $\cdots$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Unit squares of side 1 cm are joined to build bigger and bigger squares: figure 1 is a 1x1 square, figure 2 a 2x2 square, figure 3 a 3x3 square, and so on. I must find the perimeter of the eighth figure.

Givens

Each unit square has a side length of 1 cm.
Figure 1 is a 1x1 square (1 unit square).
Figure 2 is a 2x2 square (4 unit squares).
Figure 3 is a 3x3 square (9 unit squares).
The pattern continues the same way, so figure n is an n-by-n square.

Unknowns

The perimeter of the eighth figure, in cm.

Constraints

Figure n is a square whose side is n unit squares, i.e. n cm long.
Perimeter means the total length around the outside.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Each figure is a square, so I first solve the easier related problem of finding each figure's side length, spot the pattern that figure n has side n cm, and then apply the square-perimeter rule (4 times the side) to figure 8.

3 · Execute3 carry out the plan

1Find the side-length pattern

#5 Look for a Pattern 4.OA.C.5

Figure 1 has side 1 cm, figure 2 has side 2 cm, figure 3 has side 3 cm. The side of figure n equals n cm, so the eighth figure has side 8 cm.

\text{side of figure } n = n \text{ cm} \Rightarrow \text{figure 8 side} = 8 \text{ cm}

Spotting that the side grows by 1 cm each step is the Grade 4 idea of generating a shape pattern from a rule.

2Use the perimeter of each small case

#9 Solve an Easier Related Problem 3.MD.D.8

Check the rule on easy cases: figure 1 perimeter = 4 x 1 = 4 cm, figure 2 = 4 x 2 = 8 cm, figure 3 = 4 x 3 = 12 cm. The perimeter is always 4 times the side.

4 \times 1 = 4, \quad 4 \times 2 = 8, \quad 4 \times 3 = 12

Working the small figures first shows the simple 'four equal sides' perimeter rule before jumping to the target figure.

3Compute figure 8's perimeter

#1 Draw a Diagram 4.MD.A.3

The eighth figure is a 8 cm by 8 cm square, so its perimeter is 4 times 8 cm.

4 \times 8 = 32

Multiplying side by 4 for a square's perimeter is a basic Grade 4 rectangle-perimeter formula.

Answer: 32 cm

4 · Reviewdoes it hold up?

The perimeters grow 4, 8, 12, ..., increasing by 4 cm each step, which fits a side that grows by 1 cm each step; 32 cm for figure 8 continues this pattern exactly (4 x 8 = 32). Only the outer edge counts, so the inner grid lines correctly do not add to the perimeter.

Another way: Evaluate finite differences (tool 14): the perimeters 4, 8, 12, ... rise by a constant 4 each time, so the eighth term is 4 x 8 = 32 cm, confirming the result.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing that figure n is an n-by-n square and extending the pattern to figure 8.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding each square figure's perimeter from its side length.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Applying perimeter = 4 x side to the 8 cm square.

💡Takeaway. Figure number tells you the side in cm, and a square's perimeter is just 4 times its side - so figure 8 is 4 x 8 = 32 cm!

Variant 5 medium

Unit squares with a side length of $1\,\text{cm}$ are joined together without gaps or overlaps following the pattern shown below. What is the perimeter of the tenth figure, in $\text{cm}$ ?

Figure description: Small squares, each with a side length of $1\,\text{cm}$ , are joined edge to edge to build larger and larger square shapes. The first figure is a single small square ( $1\times1$ ), the second figure is a $2\times2$ square made of $4$ small squares, and the third figure is a $3\times3$ square made of $9$ small squares; the pattern continues in the same way ( $\cdots$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Unit squares of side 1 cm are joined to build bigger and bigger squares: figure 1 is a 1x1 square, figure 2 a 2x2 square, figure 3 a 3x3 square, and so on. I must find the perimeter of the tenth figure.

Givens

Each unit square has a side length of 1 cm.
Figure 1 is a 1x1 square (1 unit square).
Figure 2 is a 2x2 square (4 unit squares).
Figure 3 is a 3x3 square (9 unit squares).
The pattern continues the same way, so figure n is an n-by-n square.

Unknowns

The perimeter of the tenth figure, in cm.

Constraints

Figure n is a square whose side is n unit squares, i.e. n cm long.
Perimeter means the total length around the outside.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Each figure is a square, so I first solve the easier related problem of finding each figure's side length, spot the pattern that figure n has side n cm, and then apply the square-perimeter rule (4 times the side) to figure 10.

3 · Execute3 carry out the plan

1Find the side-length pattern

#5 Look for a Pattern 4.OA.C.5

Figure 1 has side 1 cm, figure 2 has side 2 cm, figure 3 has side 3 cm. The side of figure n equals n cm, so the tenth figure has side 10 cm.

\text{side of figure } n = n \text{ cm} \Rightarrow \text{figure 10 side} = 10 \text{ cm}

Spotting that the side grows by 1 cm each step is the Grade 4 idea of generating a shape pattern from a rule.

2Use the perimeter of each small case

#9 Solve an Easier Related Problem 3.MD.D.8

Check the rule on easy cases: figure 1 perimeter = 4 x 1 = 4 cm, figure 2 = 4 x 2 = 8 cm, figure 3 = 4 x 3 = 12 cm. The perimeter is always 4 times the side.

4 \times 1 = 4, \quad 4 \times 2 = 8, \quad 4 \times 3 = 12

Working the small figures first shows the simple 'four equal sides' perimeter rule before jumping to the target figure.

3Compute figure 10's perimeter

#1 Draw a Diagram 4.MD.A.3

The tenth figure is a 10 cm by 10 cm square, so its perimeter is 4 times 10 cm.

4 \times 10 = 40

Multiplying side by 4 for a square's perimeter is a basic Grade 4 rectangle-perimeter formula.

Answer: 40 cm

4 · Reviewdoes it hold up?

The perimeters grow 4, 8, 12, ..., increasing by 4 cm each step, which fits a side that grows by 1 cm each step; 40 cm for figure 10 continues this pattern exactly (4 x 10 = 40). Only the outer edge counts, so the inner grid lines correctly do not add to the perimeter.

Another way: Evaluate finite differences (tool 14): the perimeters 4, 8, 12, ... rise by a constant 4 each time, so the tenth term is 4 x 10 = 40 cm, confirming the result.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing that figure n is an n-by-n square and extending the pattern to figure 10.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding each square figure's perimeter from its side length.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Applying perimeter = 4 x side to the 10 cm square.

💡Takeaway. Figure number tells you the side in cm, and a square's perimeter is just 4 times its side - so figure 10 is 4 x 10 = 40 cm!

Variant 6 medium

Unit squares with a side length of $1\,\text{cm}$ are joined together without gaps or overlaps following the pattern shown below. What is the perimeter of the fourth figure, in $\text{cm}$ ?

Figure description: Small squares, each with a side length of $1\,\text{cm}$ , are joined edge to edge to build larger and larger square shapes. The first figure is a single small square ( $1\times1$ ), the second figure is a $2\times2$ square made of $4$ small squares, and the third figure is a $3\times3$ square made of $9$ small squares; the pattern continues in the same way ( $\cdots$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Unit squares of side 1 cm are joined to build bigger and bigger squares: figure 1 is a 1x1 square, figure 2 a 2x2 square, figure 3 a 3x3 square, and so on. I must find the perimeter of the fourth figure.

Givens

Each unit square has a side length of 1 cm.
Figure 1 is a 1x1 square (1 unit square).
Figure 2 is a 2x2 square (4 unit squares).
Figure 3 is a 3x3 square (9 unit squares).
The pattern continues the same way, so figure n is an n-by-n square.

Unknowns

The perimeter of the fourth figure, in cm.

Constraints

Figure n is a square whose side is n unit squares, i.e. n cm long.
Perimeter means the total length around the outside.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Each figure is a square, so I first solve the easier related problem of finding each figure's side length, spot the pattern that figure n has side n cm, and then apply the square-perimeter rule (4 times the side) to figure 4.

3 · Execute3 carry out the plan

1Find the side-length pattern

#5 Look for a Pattern 4.OA.C.5

Figure 1 has side 1 cm, figure 2 has side 2 cm, figure 3 has side 3 cm. The side of figure n equals n cm, so the fourth figure has side 4 cm.

\text{side of figure } n = n \text{ cm} \Rightarrow \text{figure 4 side} = 4 \text{ cm}

Spotting that the side grows by 1 cm each step is the Grade 4 idea of generating a shape pattern from a rule.

2Use the perimeter of each small case

#9 Solve an Easier Related Problem 3.MD.D.8

Check the rule on easy cases: figure 1 perimeter = 4 x 1 = 4 cm, figure 2 = 4 x 2 = 8 cm, figure 3 = 4 x 3 = 12 cm. The perimeter is always 4 times the side.

4 \times 1 = 4, \quad 4 \times 2 = 8, \quad 4 \times 3 = 12

Working the small figures first shows the simple 'four equal sides' perimeter rule before jumping to the target figure.

3Compute figure 4's perimeter

#1 Draw a Diagram 4.MD.A.3

The fourth figure is a 4 cm by 4 cm square, so its perimeter is 4 times 4 cm.

4 \times 4 = 16

Multiplying side by 4 for a square's perimeter is a basic Grade 4 rectangle-perimeter formula.

Answer: 16 cm

4 · Reviewdoes it hold up?

The perimeters grow 4, 8, 12, ..., increasing by 4 cm each step, which fits a side that grows by 1 cm each step; 16 cm for figure 4 continues this pattern exactly (4 x 4 = 16). Only the outer edge counts, so the inner grid lines correctly do not add to the perimeter.

Another way: Evaluate finite differences (tool 14): the perimeters 4, 8, 12, ... rise by a constant 4 each time, so the fourth term is 4 x 4 = 16 cm, confirming the result.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing that figure n is an n-by-n square and extending the pattern to figure 4.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding each square figure's perimeter from its side length.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Applying perimeter = 4 x side to the 4 cm square.

💡Takeaway. Figure number tells you the side in cm, and a square's perimeter is just 4 times its side - so figure 4 is 4 x 4 = 16 cm!

Variant 7 medium

Unit squares with a side length of $1\,\text{cm}$ are joined together without gaps or overlaps following the pattern shown below. What is the perimeter of the seventh figure, in $\text{cm}$ ?

Figure description: Small squares, each with a side length of $1\,\text{cm}$ , are joined edge to edge to build larger and larger square shapes. The first figure is a single small square ( $1\times1$ ), the second figure is a $2\times2$ square made of $4$ small squares, and the third figure is a $3\times3$ square made of $9$ small squares; the pattern continues in the same way ( $\cdots$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Unit squares of side 1 cm are joined to build bigger and bigger squares: figure 1 is a 1x1 square, figure 2 a 2x2 square, figure 3 a 3x3 square, and so on. I must find the perimeter of the seventh figure.

Givens

Each unit square has a side length of 1 cm.
Figure 1 is a 1x1 square (1 unit square).
Figure 2 is a 2x2 square (4 unit squares).
Figure 3 is a 3x3 square (9 unit squares).
The pattern continues the same way, so figure n is an n-by-n square.

Unknowns

The perimeter of the seventh figure, in cm.

Constraints

Figure n is a square whose side is n unit squares, i.e. n cm long.
Perimeter means the total length around the outside.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Each figure is a square, so I first solve the easier related problem of finding each figure's side length, spot the pattern that figure n has side n cm, and then apply the square-perimeter rule (4 times the side) to figure 7.

3 · Execute3 carry out the plan

1Find the side-length pattern

#5 Look for a Pattern 4.OA.C.5

Figure 1 has side 1 cm, figure 2 has side 2 cm, figure 3 has side 3 cm. The side of figure n equals n cm, so the seventh figure has side 7 cm.

\text{side of figure } n = n \text{ cm} \Rightarrow \text{figure 7 side} = 7 \text{ cm}

Spotting that the side grows by 1 cm each step is the Grade 4 idea of generating a shape pattern from a rule.

2Use the perimeter of each small case

#9 Solve an Easier Related Problem 3.MD.D.8

Check the rule on easy cases: figure 1 perimeter = 4 x 1 = 4 cm, figure 2 = 4 x 2 = 8 cm, figure 3 = 4 x 3 = 12 cm. The perimeter is always 4 times the side.

4 \times 1 = 4, \quad 4 \times 2 = 8, \quad 4 \times 3 = 12

Working the small figures first shows the simple 'four equal sides' perimeter rule before jumping to the target figure.

3Compute figure 7's perimeter

#1 Draw a Diagram 4.MD.A.3

The seventh figure is a 7 cm by 7 cm square, so its perimeter is 4 times 7 cm.

4 \times 7 = 28

Multiplying side by 4 for a square's perimeter is a basic Grade 4 rectangle-perimeter formula.

Answer: 28 cm

4 · Reviewdoes it hold up?

The perimeters grow 4, 8, 12, ..., increasing by 4 cm each step, which fits a side that grows by 1 cm each step; 28 cm for figure 7 continues this pattern exactly (4 x 7 = 28). Only the outer edge counts, so the inner grid lines correctly do not add to the perimeter.

Another way: Evaluate finite differences (tool 14): the perimeters 4, 8, 12, ... rise by a constant 4 each time, so the seventh term is 4 x 7 = 28 cm, confirming the result.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing that figure n is an n-by-n square and extending the pattern to figure 7.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding each square figure's perimeter from its side length.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Applying perimeter = 4 x side to the 7 cm square.

💡Takeaway. Figure number tells you the side in cm, and a square's perimeter is just 4 times its side - so figure 7 is 4 x 7 = 28 cm!

Variant 8 medium

Unit squares with a side length of $1\,\text{cm}$ are joined together without gaps or overlaps following the pattern shown below. What is the perimeter of the fifth figure, in $\text{cm}$ ?

Figure description: Small squares, each with a side length of $1\,\text{cm}$ , are joined edge to edge to build larger and larger square shapes. The first figure is a single small square ( $1\times1$ ), the second figure is a $2\times2$ square made of $4$ small squares, and the third figure is a $3\times3$ square made of $9$ small squares; the pattern continues in the same way ( $\cdots$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Unit squares of side 1 cm are joined to build bigger and bigger squares: figure 1 is a 1x1 square, figure 2 a 2x2 square, figure 3 a 3x3 square, and so on. I must find the perimeter of the fifth figure.

Givens

Each unit square has a side length of 1 cm.
Figure 1 is a 1x1 square (1 unit square).
Figure 2 is a 2x2 square (4 unit squares).
Figure 3 is a 3x3 square (9 unit squares).
The pattern continues the same way, so figure n is an n-by-n square.

Unknowns

The perimeter of the fifth figure, in cm.

Constraints

Figure n is a square whose side is n unit squares, i.e. n cm long.
Perimeter means the total length around the outside.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Each figure is a square, so I first solve the easier related problem of finding each figure's side length, spot the pattern that figure n has side n cm, and then apply the square-perimeter rule (4 times the side) to figure 5.

3 · Execute3 carry out the plan

1Find the side-length pattern

#5 Look for a Pattern 4.OA.C.5

Figure 1 has side 1 cm, figure 2 has side 2 cm, figure 3 has side 3 cm. The side of figure n equals n cm, so the fifth figure has side 5 cm.

\text{side of figure } n = n \text{ cm} \Rightarrow \text{figure 5 side} = 5 \text{ cm}

Spotting that the side grows by 1 cm each step is the Grade 4 idea of generating a shape pattern from a rule.

2Use the perimeter of each small case

#9 Solve an Easier Related Problem 3.MD.D.8

Check the rule on easy cases: figure 1 perimeter = 4 x 1 = 4 cm, figure 2 = 4 x 2 = 8 cm, figure 3 = 4 x 3 = 12 cm. The perimeter is always 4 times the side.

4 \times 1 = 4, \quad 4 \times 2 = 8, \quad 4 \times 3 = 12

Working the small figures first shows the simple 'four equal sides' perimeter rule before jumping to the target figure.

3Compute figure 5's perimeter

#1 Draw a Diagram 4.MD.A.3

The fifth figure is a 5 cm by 5 cm square, so its perimeter is 4 times 5 cm.

4 \times 5 = 20

Multiplying side by 4 for a square's perimeter is a basic Grade 4 rectangle-perimeter formula.

Answer: 20 cm

4 · Reviewdoes it hold up?

The perimeters grow 4, 8, 12, ..., increasing by 4 cm each step, which fits a side that grows by 1 cm each step; 20 cm for figure 5 continues this pattern exactly (4 x 5 = 20). Only the outer edge counts, so the inner grid lines correctly do not add to the perimeter.

Another way: Evaluate finite differences (tool 14): the perimeters 4, 8, 12, ... rise by a constant 4 each time, so the fifth term is 4 x 5 = 20 cm, confirming the result.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing that figure n is an n-by-n square and extending the pattern to figure 5.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding each square figure's perimeter from its side length.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Applying perimeter = 4 x side to the 5 cm square.

💡Takeaway. Figure number tells you the side in cm, and a square's perimeter is just 4 times its side - so figure 5 is 4 x 5 = 20 cm!

Perimeter of the nth figure by rule · 8 practice problems

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