Practice · Side lengths from overlapping rectangles · Sensim Math

Variant 1 easy

Rectangle A and square B overlap as shown below. The perimeter of rectangle A is $18\,\text{cm}$ . 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 $5\,\text{cm}$ . Inside the overlap, the part belonging to rectangle A has a vertical length of $2\,\text{cm}$ , and the part belonging to square B has a vertical length of $3\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

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 18 cm and its top side is 5 cm. Inside the overlap, the vertical part belonging to A is 2 cm and the vertical part belonging to B is 3 cm. I must find the side length of square B.

Givens

Rectangle A has perimeter 18 cm.
The top side of rectangle A is 5 cm, so its width is 5 cm.
Along the overlap, A's vertical part is 2 cm.
Along B's left side, the part below A's bottom edge is 3 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 2 cm overlap part and the 3 cm part below A together make one full side of B.

2 · Planchoose the strategy

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

The width plus the perimeter let 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 top part inside the overlap and a bottom part below A, and adding those two pieces gives B's full side.

3 · Execute3 carry out the plan

1Recover rectangle A's height

#11 Work Backwards 3.MD.D.8

A's perimeter is 18 cm and its width is 5 cm. Working backwards, the two widths use 5 + 5 = 10 cm, leaving 18 - 10 = 8 cm for the two heights, so each height is 4 cm. (This confirms the figure but is not needed for B.)

18 - 2 \times 5 = 8, \quad 8 \div 2 = 4

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

2See how B's side is split

#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, 2 cm) and a bottom piece (below A, 3 cm).

\text{side of } B = 2 + 3

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

3Add the two pieces

#7 Identify Subproblems 4.MD.A.3

Because B is a square, the side made of the 2 cm and 3 cm pieces is one full side length of B.

2 + 3 = 5

Adding the overlap part and the protruding part to get the whole side is straightforward addition.

Answer: 5 cm

4 · Reviewdoes it hold up?

B's side 5 cm is longer than the 3 cm bottom piece and the 2 cm overlap piece, which it must be since it contains both. It is comparable in size to rectangle A's 5 cm and 4 cm sides, so two figures of this scale overlapping at a corner is sensible.

Another way: Convert to a tiny equation (tool 13): let s be B's side; the side equals overlap (2) plus the protruding part (3), so s = 2 + 3 = 5 cm, matching the diagram reasoning.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Working backwards from A's 18 cm perimeter and 5 cm width to its 4 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 2 cm and 3 cm pieces.

💡Takeaway. Find where one shape's edge cuts across the other, then add the two pieces of that side - it's just 2 + 3!

Variant 2 easy

Rectangle A and square B overlap as shown below. The perimeter of rectangle A is $22\,\text{cm}$ . 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 $6\,\text{cm}$ . Inside the overlap, the part belonging to rectangle A has a vertical length of $4\,\text{cm}$ , and the part belonging to square B has a vertical length of $4\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

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 22 cm and its top side is 6 cm. Inside the overlap, the vertical part belonging to A is 4 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 22 cm.
The top side of rectangle A is 6 cm, so its width is 6 cm.
Along the overlap, A's vertical part is 4 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 4 cm overlap part and the 4 cm part below A together make one full side of B.

2 · Planchoose the strategy

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

The width plus the perimeter let 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 top part inside the overlap and a bottom part below A, and adding those two pieces gives B's full side.

3 · Execute3 carry out the plan

1Recover rectangle A's height

#11 Work Backwards 3.MD.D.8

A's perimeter is 22 cm and its width is 6 cm. Working backwards, the two widths use 6 + 6 = 12 cm, leaving 22 - 12 = 10 cm for the two heights, so each height is 5 cm. (This confirms the figure but is not needed for B.)

22 - 2 \times 6 = 10, \quad 10 \div 2 = 5

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

2See how B's side is split

#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, 4 cm) and a bottom piece (below A, 4 cm).

\text{side of } B = 4 + 4

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

3Add the two pieces

#7 Identify Subproblems 4.MD.A.3

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

4 + 4 = 8

Adding the overlap part and the protruding part to get the whole side is straightforward addition.

Answer: 8 cm

4 · Reviewdoes it hold up?

B's side 8 cm is longer than the 4 cm bottom piece and the 4 cm overlap piece, which it must be since it contains both. It is comparable in size to rectangle A's 6 cm and 5 cm sides, so two figures of this scale overlapping at a corner is sensible.

Another way: Convert to a tiny equation (tool 13): let s be B's side; the side equals overlap (4) plus the protruding part (4), so s = 4 + 4 = 8 cm, matching the diagram reasoning.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Working backwards from A's 22 cm perimeter and 6 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 4 cm and 4 cm pieces.

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

Variant 3 medium

Rectangle A and square B overlap as shown below. The perimeter of rectangle A is $24\,\text{cm}$ . 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 $7\,\text{cm}$ . Inside the overlap, the part belonging to rectangle A has a vertical length of $1\,\text{cm}$ , and the part belonging to square B has a vertical length of $8\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

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 24 cm and its top side is 7 cm. Inside the overlap, the vertical part belonging to A is 1 cm and the vertical part belonging to B is 8 cm. I must find the side length of square B.

Givens

Rectangle A has perimeter 24 cm.
The top side of rectangle A is 7 cm, so its width is 7 cm.
Along the overlap, A's vertical part is 1 cm.
Along B's left side, the part below A's bottom edge is 8 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 1 cm overlap part and the 8 cm part below A together make one full side of B.

2 · Planchoose the strategy

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

The width plus the perimeter let 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 top part inside the overlap and a bottom part below A, and adding those two pieces gives B's full side.

3 · Execute3 carry out the plan

1Recover rectangle A's height

#11 Work Backwards 3.MD.D.8

A's perimeter is 24 cm and its width is 7 cm. Working backwards, the two widths use 7 + 7 = 14 cm, leaving 24 - 14 = 10 cm for the two heights, so each height is 5 cm. (This confirms the figure but is not needed for B.)

24 - 2 \times 7 = 10, \quad 10 \div 2 = 5

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

2See how B's side is split

#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, 1 cm) and a bottom piece (below A, 8 cm).

\text{side of } B = 1 + 8

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

3Add the two pieces

#7 Identify Subproblems 4.MD.A.3

Because B is a square, the side made of the 1 cm and 8 cm pieces is one full side length of B.

1 + 8 = 9

Adding the overlap part and the protruding part to get the whole side is straightforward addition.

Answer: 9 cm

4 · Reviewdoes it hold up?

B's side 9 cm is longer than the 8 cm bottom piece and the 1 cm overlap piece, which it must be since it contains both. It is comparable in size to rectangle A's 7 cm and 5 cm sides, so two figures of this scale overlapping at a corner is sensible.

Another way: Convert to a tiny equation (tool 13): let s be B's side; the side equals overlap (1) plus the protruding part (8), so s = 1 + 8 = 9 cm, matching the diagram reasoning.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Working backwards from A's 24 cm perimeter and 7 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 1 cm and 8 cm pieces.

💡Takeaway. Find where one shape's edge cuts across the other, then add the two pieces of that side - it's just 1 + 8!

Variant 4 medium

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 $4\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

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 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.

2 · Planchoose the strategy

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

The width plus the perimeter let 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 top part inside the overlap and a bottom part below A, and adding those two pieces gives B's full side.

3 · Execute3 carry out the plan

1Recover rectangle A's height

#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 'work backwards' with perimeter.

2See how B's side is split

#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.

3Add the two pieces

#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 protruding part to get the whole side is straightforward addition.

Answer: 7 cm

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 5 medium

Rectangle A and square B overlap as shown below. The perimeter of rectangle A is $30\,\text{cm}$ . 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 $9\,\text{cm}$ . Inside the overlap, the part belonging to rectangle A has a vertical length of $5\,\text{cm}$ , and the part belonging to square B has a vertical length of $2\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

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 30 cm and its top side is 9 cm. Inside the overlap, the vertical part belonging to A is 5 cm and the vertical part belonging to B is 2 cm. I must find the side length of square B.

Givens

Rectangle A has perimeter 30 cm.
The top side of rectangle A is 9 cm, so its width is 9 cm.
Along the overlap, A's vertical part is 5 cm.
Along B's left side, the part below A's bottom edge is 2 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 5 cm overlap part and the 2 cm part below A together make one full side of B.

2 · Planchoose the strategy

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

The width plus the perimeter let 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 top part inside the overlap and a bottom part below A, and adding those two pieces gives B's full side.

3 · Execute3 carry out the plan

1Recover rectangle A's height

#11 Work Backwards 3.MD.D.8

A's perimeter is 30 cm and its width is 9 cm. Working backwards, the two widths use 9 + 9 = 18 cm, leaving 30 - 18 = 12 cm for the two heights, so each height is 6 cm. (This confirms the figure but is not needed for B.)

30 - 2 \times 9 = 12, \quad 12 \div 2 = 6

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

2See how B's side is split

#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, 5 cm) and a bottom piece (below A, 2 cm).

\text{side of } B = 5 + 2

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

3Add the two pieces

#7 Identify Subproblems 4.MD.A.3

Because B is a square, the side made of the 5 cm and 2 cm pieces is one full side length of B.

5 + 2 = 7

Adding the overlap part and the protruding part to get the whole side is straightforward addition.

Answer: 7 cm

4 · Reviewdoes it hold up?

B's side 7 cm is longer than the 2 cm bottom piece and the 5 cm overlap piece, which it must be since it contains both. It is comparable in size to rectangle A's 9 cm and 6 cm sides, so two figures of this scale overlapping at a corner is sensible.

Another way: Convert to a tiny equation (tool 13): let s be B's side; the side equals overlap (5) plus the protruding part (2), so s = 5 + 2 = 7 cm, matching the diagram reasoning.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Working backwards from A's 30 cm perimeter and 9 cm width to its 6 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 5 cm and 2 cm pieces.

💡Takeaway. Find where one shape's edge cuts across the other, then add the two pieces of that side - it's just 5 + 2!

Variant 6 medium

Rectangle A and square B overlap as shown below. The perimeter of rectangle A is $32\,\text{cm}$ . 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 $10\,\text{cm}$ . Inside the overlap, the part belonging to rectangle A has a vertical length of $2\,\text{cm}$ , and the part belonging to square B has a vertical length of $5\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

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 32 cm and its top side is 10 cm. Inside the overlap, the vertical part belonging to A is 2 cm and the vertical part belonging to B is 5 cm. I must find the side length of square B.

Givens

Rectangle A has perimeter 32 cm.
The top side of rectangle A is 10 cm, so its width is 10 cm.
Along the overlap, A's vertical part is 2 cm.
Along B's left side, the part below A's bottom edge is 5 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 2 cm overlap part and the 5 cm part below A together make one full side of B.

2 · Planchoose the strategy

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

The width plus the perimeter let 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 top part inside the overlap and a bottom part below A, and adding those two pieces gives B's full side.

3 · Execute3 carry out the plan

1Recover rectangle A's height

#11 Work Backwards 3.MD.D.8

A's perimeter is 32 cm and its width is 10 cm. Working backwards, the two widths use 10 + 10 = 20 cm, leaving 32 - 20 = 12 cm for the two heights, so each height is 6 cm. (This confirms the figure but is not needed for B.)

32 - 2 \times 10 = 12, \quad 12 \div 2 = 6

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

2See how B's side is split

#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, 2 cm) and a bottom piece (below A, 5 cm).

\text{side of } B = 2 + 5

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

3Add the two pieces

#7 Identify Subproblems 4.MD.A.3

Because B is a square, the side made of the 2 cm and 5 cm pieces is one full side length of B.

2 + 5 = 7

Adding the overlap part and the protruding part to get the whole side is straightforward addition.

Answer: 7 cm

4 · Reviewdoes it hold up?

B's side 7 cm is longer than the 5 cm bottom piece and the 2 cm overlap piece, which it must be since it contains both. It is comparable in size to rectangle A's 10 cm and 6 cm sides, so two figures of this scale overlapping at a corner is sensible.

Another way: Convert to a tiny equation (tool 13): let s be B's side; the side equals overlap (2) plus the protruding part (5), so s = 2 + 5 = 7 cm, matching the diagram reasoning.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Working backwards from A's 32 cm perimeter and 10 cm width to its 6 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 2 cm and 5 cm pieces.

💡Takeaway. Find where one shape's edge cuts across the other, then add the two pieces of that side - it's just 2 + 5!

Variant 7 hard

Rectangle A and square B overlap as shown below. The perimeter of rectangle A is $40\,\text{cm}$ . 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 $12\,\text{cm}$ . 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 $6\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

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 40 cm and its top side is 12 cm. Inside the overlap, the vertical part belonging to A is 3 cm and the vertical part belonging to B is 6 cm. I must find the side length of square B.

Givens

Rectangle A has perimeter 40 cm.
The top side of rectangle A is 12 cm, so its width is 12 cm.
Along the overlap, A's vertical part is 3 cm.
Along B's left side, the part below A's bottom edge is 6 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 6 cm part below A together make one full side of B.

2 · Planchoose the strategy

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

The width plus the perimeter let 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 top part inside the overlap and a bottom part below A, and adding those two pieces gives B's full side.

3 · Execute3 carry out the plan

1Recover rectangle A's height

#11 Work Backwards 3.MD.D.8

A's perimeter is 40 cm and its width is 12 cm. Working backwards, the two widths use 12 + 12 = 24 cm, leaving 40 - 24 = 16 cm for the two heights, so each height is 8 cm. (This confirms the figure but is not needed for B.)

40 - 2 \times 12 = 16, \quad 16 \div 2 = 8

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

2See how B's side is split

#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, 6 cm).

\text{side of } B = 3 + 6

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

3Add the two pieces

#7 Identify Subproblems 4.MD.A.3

Because B is a square, the side made of the 3 cm and 6 cm pieces is one full side length of B.

3 + 6 = 9

Adding the overlap part and the protruding part to get the whole side is straightforward addition.

Answer: 9 cm

4 · Reviewdoes it hold up?

B's side 9 cm is longer than the 6 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 12 cm and 8 cm sides, so two figures of this scale overlapping at a corner is sensible.

Another way: Convert to a tiny equation (tool 13): let s be B's side; the side equals overlap (3) plus the protruding part (6), so s = 3 + 6 = 9 cm, matching the diagram reasoning.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Working backwards from A's 40 cm perimeter and 12 cm width to its 8 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 6 cm pieces.

💡Takeaway. Find where one shape's edge cuts across the other, then add the two pieces of that side - it's just 3 + 6!

Variant 8 hard

Rectangle A and square B overlap as shown below. The perimeter of rectangle A is $50\,\text{cm}$ . 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 $15\,\text{cm}$ . Inside the overlap, the part belonging to rectangle A has a vertical length of $6\,\text{cm}$ , and the part belonging to square B has a vertical length of $4\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

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 50 cm and its top side is 15 cm. Inside the overlap, the vertical part belonging to A is 6 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 50 cm.
The top side of rectangle A is 15 cm, so its width is 15 cm.
Along the overlap, A's vertical part is 6 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 6 cm overlap part and the 4 cm part below A together make one full side of B.

2 · Planchoose the strategy

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

The width plus the perimeter let 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 top part inside the overlap and a bottom part below A, and adding those two pieces gives B's full side.

3 · Execute3 carry out the plan

1Recover rectangle A's height

#11 Work Backwards 3.MD.D.8

A's perimeter is 50 cm and its width is 15 cm. Working backwards, the two widths use 15 + 15 = 30 cm, leaving 50 - 30 = 20 cm for the two heights, so each height is 10 cm. (This confirms the figure but is not needed for B.)

50 - 2 \times 15 = 20, \quad 20 \div 2 = 10

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

2See how B's side is split

#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, 6 cm) and a bottom piece (below A, 4 cm).

\text{side of } B = 6 + 4

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

3Add the two pieces

#7 Identify Subproblems 4.MD.A.3

Because B is a square, the side made of the 6 cm and 4 cm pieces is one full side length of B.

6 + 4 = 10

Adding the overlap part and the protruding part to get the whole side is straightforward addition.

Answer: 10 cm

4 · Reviewdoes it hold up?

B's side 10 cm is longer than the 4 cm bottom piece and the 6 cm overlap piece, which it must be since it contains both. It is comparable in size to rectangle A's 15 cm and 10 cm sides, so two figures of this scale overlapping at a corner is sensible.

Another way: Convert to a tiny equation (tool 13): let s be B's side; the side equals overlap (6) plus the protruding part (4), so s = 6 + 4 = 10 cm, matching the diagram reasoning.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Working backwards from A's 50 cm perimeter and 15 cm width to its 10 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 6 cm and 4 cm pieces.

💡Takeaway. Find where one shape's edge cuts across the other, then add the two pieces of that side - it's just 6 + 4!

Side lengths from overlapping rectangles · 8 practice problems

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