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

Perimeter by tracing every side

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.

Measurement & data

Your answer

How to solve

Strategy Draw a Diagram — 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.

1STEP 1

Set up the outline on a grid

Top-left corner at the origin: the top row spans 27 cm across and 6 cm down, and the left column reaches 12 cm tall.

3 × 9 = 27 cm (top width), 2 × 6 = 12 cm (left height)

2STEP 2

Add the horizontal edges

Horizontal pieces: the top (27 cm), the step (27 - 9 = 18 cm), and the bottom (9 cm).

27 + 18 + 9 = 54

3STEP 3

Add the vertical edges

Vertical pieces: the top row's right side (6 cm), the step drop (6 cm), and the tall left side (12 cm).

6 + 6 + 12 = 24

4STEP 4

Total the perimeter

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

54 + 24 = 78

Answer

78 cm

54 + 24 = 78

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.

Takeaway

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

Set up the outline on a grid
Add the horizontal edges
Add the vertical edges
Total the perimeter

Where next?

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▶ Practice — 10 problems

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