Side lengths from overlapping rectangles

3.MD.D.84.MD.A.33.OA.D.8 · adapt · grade 4

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

Rectangle A and square B overlap as shown below. The perimeter of rectangle A is $26\,\text{cm}$ . What is the side length of square B, in $\text{cm}$ ?

Figure description: Rectangle A sits at the upper left and square B sits at the lower right, overlapping at one corner so that the overlap is a small shaded rectangle. The top side of rectangle A is labeled $8\,\text{cm}$ . Inside the overlap, the part belonging to rectangle A has a vertical length of $3\,\text{cm}$ , and the part belonging to square B has a vertical length of $4\,\text{cm}$ .

Show solution

Understand

Rectangle A (top-left) and square B (bottom-right) overlap at one corner, and the overlap is a small shaded rectangle. Rectangle A's perimeter is 26 cm and its top side is 8 cm. Inside the overlap, the vertical part belonging to A is 3 cm and the vertical part belonging to B is 4 cm. I must find the side length of square B.

Givens

Rectangle A has perimeter 26 cm.
The top side of rectangle A is 8 cm, so its width is 8 cm.
Along the overlap, A's vertical part (from where B's top edge crosses A down to A's bottom edge) is 3 cm.
Along B's left side, the part below A's bottom edge is 4 cm.
B is a square, so all its sides are equal.

Unknowns

The side length of square B, in cm.

Constraints

Opposite sides of a rectangle are equal; all sides of a square are equal.
B's left side runs straight down: the 3 cm overlap part and the 4 cm part below A together make one full side of B.

Plan

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

The 8 cm width plus the 26 cm perimeter lets me work backwards to A's height, but the height is not even needed for B. The real key is the diagram: B's left side is split by A's bottom edge into a 3 cm top part (inside the overlap) and a 4 cm bottom part, and adding those two subproblem pieces gives B's full side.

Execute

#11 Work Backwards 3.MD.D.8

A's perimeter is 26 cm and its width is 8 cm. Working backwards, the two widths use 8 + 8 = 16 cm, leaving 26 - 16 = 10 cm for the two heights, so each height is 5 cm. (This confirms the figure but is not needed for B.)

26 - 2 \times 8 = 10, \quad 10 \div 2 = 5

Undoing the perimeter to find a missing side is a natural Grade 3 'work backwards' with perimeter.

#1 Draw a Diagram 3.OA.D.8

B's left side runs straight down from inside rectangle A. A's bottom edge crosses it, splitting that side into a top piece (the overlap part, 3 cm) and a bottom piece (below A, 4 cm).

\text{side of } B = 3 + 4

Drawing where A's bottom edge cuts across B shows the one side broken into two labeled pieces I can simply add.

#7 Identify Subproblems 4.MD.A.3

Because B is a square, the side made of the 3 cm and 4 cm pieces is one full side length of B.

3 + 4 = 7

Adding the overlap part and the below part to get the whole side is straightforward Grade 3 addition.

Answer: 7 cm

Review

B's side 7 cm is longer than the 4 cm bottom piece and the 3 cm overlap piece, which it must be since it contains both. It is comparable in size to rectangle A's 8 cm and 5 cm sides, so two figures of this scale overlapping at a corner is sensible.

Convert to a tiny equation (tool 13): let s be B's side; the side equals overlap (3) plus the protruding part (4), so s = 3 + 4 = 7 cm, matching the diagram reasoning.

Standards · min grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Working backwards from A's 26 cm perimeter and 8 cm width to its 5 cm height.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perimeter step and the side-splitting step to reach the answer.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Reasoning about the square's equal side built from the 3 cm and 4 cm pieces.

💡 Find where one shape's edge cuts across the other, then add the two pieces of that side - it's just 3 + 4!