Patterns & Reasoning

Depth · one concept from first principles · step 2 of 4

Perimeter of the nth figure by rule

Unit squares with a side length of 1 cm1\,\text{cm} are joined together without gaps or overlaps following the pattern shown below. What is the perimeter of the fourth figure, in cm\text{cm}?

Figure description: Small squares, each with a side length of 1 cm1\,\text{cm}, are joined edge to edge to build larger and larger square shapes. The first figure is a single small square (1×11\times1), the second figure is a 2×22\times2 square made of 44 small squares, and the third figure is a 3×33\times3 square made of 99 small squares; the pattern continues in the same way (⋯\cdots).

Figure 1 Figure 2 Figure 3
OperationsMeasurement & data
Your answer
cm
How to solve
Strategy Look for a Pattern — 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.
1STEP 1

Find the side-length pattern

Figures 1, 2, 3 have sides 1, 2, 3 cm - figure n's side is n cm, so figure 4 has side 4 cm.

side of figure n = n cm → figure 4 side = 4 cm
2STEP 2

Use the perimeter of each small case

Check small cases: figure 1 = 4 cm, figure 2 = 8 cm, figure 3 = 12 cm - perimeter is always 4 times the side.

4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12
3STEP 3

Compute figure 4's perimeter

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

4 × 4 = 16
Answer
16 cm
4 × 4 = 16
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.
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!

  • Find the side-length pattern
  • Use the perimeter of each small case
  • Compute figure 4's perimeter
Where next?
Another one like thissuggested

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▶ Practice — 8 problems

The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Geometric Series Share, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →