Perimeter of the nth figure by rule

4.OA.C.53.MD.D.83.OA.D.9 · adapt · grade 4

Archetype: Generalize a Growing Pattern into a Rule · step in a 12-type progression

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 ( $\cdots$ ).

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Understand

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.

Plan

#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.

Execute

#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.

#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 figure 4.

#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

Review

The perimeters grow 4, 8, 12, 16, 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. Only the outer edge counts, so the many inner grid lines correctly do not add to the perimeter.

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

Standards · min 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.

💡 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!