Perimeter by tracing every side

3.MD.D.84.MD.A.3 · adapt · grade 4

Archetype: Perimeter by Tracing Every Side · step in a 11-type progression

Four identical rectangles, each $9\,\text{cm}$ long and $6\,\text{cm}$ tall, are joined together to make the figure shown below. What is the perimeter of this figure, in $\text{cm}$ ?

Figure description: A staircase-shaped figure is made by joining four identical rectangles (each $9\,\text{cm}$ wide and $6\,\text{cm}$ tall) edge to edge. The top row has $3$ rectangles placed side by side in a horizontal line, and one more rectangle is attached directly below the leftmost rectangle of that row. One rectangle is labeled $9\,\text{cm}$ along its horizontal side and $6\,\text{cm}$ along its vertical side.

Show solution

Understand

Four identical rectangles, each 9 cm wide and 6 cm tall, are joined into a staircase shape: three sit in a top row, and one hangs directly below the leftmost. We need the perimeter of the whole figure in centimeters.

Givens

Each rectangle is 9 cm wide and 6 cm tall.
Three rectangles form the top row, placed side by side.
A fourth rectangle is attached below the leftmost top rectangle.

Unknowns

The perimeter of the combined staircase figure.

Constraints

Rectangles are joined edge to edge with no gaps or overlaps.
All measurements are multiples of the 9 cm and 6 cm side lengths.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Sketching the figure on a grid lets us trace every outer edge in order. Breaking the boundary into separate horizontal and vertical pieces (subproblems) keeps the additions organized so no side is missed or double counted.

Execute

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

Place the figure with the top-left corner at the origin. The top row spans 3 times 9 = 27 cm across and 6 cm down. The fourth rectangle sits below the leftmost column, spanning 9 cm across and another 6 cm down, so that column reaches 12 cm tall.

3 \times 9 = 27\text{ cm (top width)}, \quad 2 \times 6 = 12\text{ cm (left height)}

Putting the picture on a grid turns 'staircase' into clear straight edges I can measure.

#7 Identify Subproblems 3.NBT.A.2

Going around, the horizontal pieces are: the full top (27 cm), the step in the middle where the top row's bottom juts out past the lower rectangle (27 - 9 = 18 cm), and the bottom of the lower rectangle (9 cm).

27 + 18 + 9 = 54

Grade 3 addition: the left-and-right edges of the outline add to 54 cm.

#7 Identify Subproblems 3.NBT.A.2

The vertical pieces are: the right side of the top row (6 cm), the short drop at the step (6 cm), and the tall left side (12 cm).

6 + 6 + 12 = 24

Grade 3 addition: the up-and-down edges of the outline add to 24 cm.

#7 Identify Subproblems 3.MD.D.8

Add the horizontal total and the vertical total to get the full distance around the figure.

54 + 24 = 78

Perimeter is just the whole trip around the edge: 54 cm across plus 24 cm up and down is 78 cm.

Answer: 78 cm

Review

A single 9 by 6 rectangle has perimeter 30 cm; four of them total 120 cm, but joining them hides several shared edges. Our 78 cm is comfortably below 120 cm and well above one rectangle's 30 cm, so the magnitude makes sense for a four-rectangle staircase.

Treat the figure as a 27 by 6 top bar plus a 9 by 6 tab. Their perimeters are 66 cm and 30 cm; subtract twice the 9 cm shared edge (2 times 9 = 18 cm): 66 + 30 - 18 = 78 cm.

Standards · min grade 4

4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Reasoning about the rectangle dimensions and the composite outline on a grid.
3.NBT.A.2 Fluently add and subtract within 1000 — Adding the horizontal and vertical edge lengths.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Combining all outer edges into the total perimeter.

💡 Trace the outline once, adding every edge as you go, and the staircase perimeter is just 78 cm!