← Solve known areas first, step by step · Area by Decomposition

Solve known areas first, step by step · 12 practice problems

4.MD.A.34.OA.A.36.G.A.1

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 12 cm

The figure at right shows a rhombus overlapped with a square whose side length is 10 cm10\,\text{cm}. The area of the rhombus is 33 times the area of the overlapping region, and the area of the square is 55 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 10 cm10\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 10 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 10 cm square. The square is 5 times the overlap and the rhombus is 3 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 10 cm.
  • Square area = 5 x overlap area.
  • Rhombus area = 3 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 10 cm, so its area is side times side. Nothing else in the figure can be measured yet.
10×10=100 cm210 \times 10 = 100\ \text{cm}^2
100 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 5 times the overlap, so the overlap is the square's area divided by 5.
100÷5=20 cm2100 \div 5 = 20\ \text{cm}^2
The shared part is 20 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 3 times the overlap, so multiply the overlap by 3.
20×3=60 cm220 \times 3 = 60\ \text{cm}^2
The rhombus is 60 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=10 cmAC = 10\ \text{cm}
AC is 10 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 60 and AC = 10 gives 10 x BD / 2 = 60.
10×BD2=60  ⇒  5×BD=60  ⇒  BD=12 cm\dfrac{10 \times BD}{2} = 60 \;\Rightarrow\; 5 \times BD = 60 \;\Rightarrow\; BD = 12\ \text{cm}
BD is 12 cm.
Answer: 12 cm
4 · Reviewdoes it hold up?

Put 12 back into the formula: 10 x 12 / 2 = 60 cm2, and that is 3 times the 20 cm2 overlap, which is 100 / 5. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 3/5 of the square, which is 3/5 x 100 = 60 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 2 easy answer: 18 cm

The figure at right shows a rhombus overlapped with a square whose side length is 12 cm12\,\text{cm}. The area of the rhombus is 33 times the area of the overlapping region, and the area of the square is 44 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 12 cm12\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 12 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 12 cm square. The square is 4 times the overlap and the rhombus is 3 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 12 cm.
  • Square area = 4 x overlap area.
  • Rhombus area = 3 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 12 cm, so its area is side times side. Nothing else in the figure can be measured yet.
12×12=144 cm212 \times 12 = 144\ \text{cm}^2
144 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 4 times the overlap, so the overlap is the square's area divided by 4.
144÷4=36 cm2144 \div 4 = 36\ \text{cm}^2
The shared part is 36 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 3 times the overlap, so multiply the overlap by 3.
36×3=108 cm236 \times 3 = 108\ \text{cm}^2
The rhombus is 108 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=12 cmAC = 12\ \text{cm}
AC is 12 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 108 and AC = 12 gives 12 x BD / 2 = 108.
12×BD2=108  ⇒  6×BD=108  ⇒  BD=18 cm\dfrac{12 \times BD}{2} = 108 \;\Rightarrow\; 6 \times BD = 108 \;\Rightarrow\; BD = 18\ \text{cm}
BD is 18 cm.
Answer: 18 cm
4 · Reviewdoes it hold up?

Put 18 back into the formula: 12 x 18 / 2 = 108 cm2, and that is 3 times the 36 cm2 overlap, which is 144 / 4. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 3/4 of the square, which is 3/4 x 144 = 108 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 3 easy answer: 20 cm

The figure at right shows a rhombus overlapped with a square whose side length is 12 cm12\,\text{cm}. The area of the rhombus is 55 times the area of the overlapping region, and the area of the square is 66 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 12 cm12\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 12 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 12 cm square. The square is 6 times the overlap and the rhombus is 5 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 12 cm.
  • Square area = 6 x overlap area.
  • Rhombus area = 5 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 12 cm, so its area is side times side. Nothing else in the figure can be measured yet.
12×12=144 cm212 \times 12 = 144\ \text{cm}^2
144 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 6 times the overlap, so the overlap is the square's area divided by 6.
144÷6=24 cm2144 \div 6 = 24\ \text{cm}^2
The shared part is 24 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 5 times the overlap, so multiply the overlap by 5.
24×5=120 cm224 \times 5 = 120\ \text{cm}^2
The rhombus is 120 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=12 cmAC = 12\ \text{cm}
AC is 12 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 120 and AC = 12 gives 12 x BD / 2 = 120.
12×BD2=120  ⇒  6×BD=120  ⇒  BD=20 cm\dfrac{12 \times BD}{2} = 120 \;\Rightarrow\; 6 \times BD = 120 \;\Rightarrow\; BD = 20\ \text{cm}
BD is 20 cm.
Answer: 20 cm
4 · Reviewdoes it hold up?

Put 20 back into the formula: 12 x 20 / 2 = 120 cm2, and that is 5 times the 24 cm2 overlap, which is 144 / 6. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 5/6 of the square, which is 5/6 x 144 = 120 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 4 easy answer: 16 cm

The figure at right shows a rhombus overlapped with a square whose side length is 12 cm12\,\text{cm}. The area of the rhombus is 22 times the area of the overlapping region, and the area of the square is 33 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 12 cm12\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 12 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 12 cm square. The square is 3 times the overlap and the rhombus is 2 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 12 cm.
  • Square area = 3 x overlap area.
  • Rhombus area = 2 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 12 cm, so its area is side times side. Nothing else in the figure can be measured yet.
12×12=144 cm212 \times 12 = 144\ \text{cm}^2
144 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 3 times the overlap, so the overlap is the square's area divided by 3.
144÷3=48 cm2144 \div 3 = 48\ \text{cm}^2
The shared part is 48 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 2 times the overlap, so multiply the overlap by 2.
48×2=96 cm248 \times 2 = 96\ \text{cm}^2
The rhombus is 96 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=12 cmAC = 12\ \text{cm}
AC is 12 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 96 and AC = 12 gives 12 x BD / 2 = 96.
12×BD2=96  ⇒  6×BD=96  ⇒  BD=16 cm\dfrac{12 \times BD}{2} = 96 \;\Rightarrow\; 6 \times BD = 96 \;\Rightarrow\; BD = 16\ \text{cm}
BD is 16 cm.
Answer: 16 cm
4 · Reviewdoes it hold up?

Put 16 back into the formula: 12 x 16 / 2 = 96 cm2, and that is 2 times the 48 cm2 overlap, which is 144 / 3. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 2/3 of the square, which is 2/3 x 144 = 96 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 5 medium answer: 16 cm

The figure at right shows a rhombus overlapped with a square whose side length is 14 cm14\,\text{cm}. The area of the rhombus is 44 times the area of the overlapping region, and the area of the square is 77 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 14 cm14\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 14 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 14 cm square. The square is 7 times the overlap and the rhombus is 4 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 14 cm.
  • Square area = 7 x overlap area.
  • Rhombus area = 4 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 14 cm, so its area is side times side. Nothing else in the figure can be measured yet.
14×14=196 cm214 \times 14 = 196\ \text{cm}^2
196 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 7 times the overlap, so the overlap is the square's area divided by 7.
196÷7=28 cm2196 \div 7 = 28\ \text{cm}^2
The shared part is 28 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 4 times the overlap, so multiply the overlap by 4.
28×4=112 cm228 \times 4 = 112\ \text{cm}^2
The rhombus is 112 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=14 cmAC = 14\ \text{cm}
AC is 14 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 112 and AC = 14 gives 14 x BD / 2 = 112.
14×BD2=112  ⇒  7×BD=112  ⇒  BD=16 cm\dfrac{14 \times BD}{2} = 112 \;\Rightarrow\; 7 \times BD = 112 \;\Rightarrow\; BD = 16\ \text{cm}
BD is 16 cm.
Answer: 16 cm
4 · Reviewdoes it hold up?

Put 16 back into the formula: 14 x 16 / 2 = 112 cm2, and that is 4 times the 28 cm2 overlap, which is 196 / 7. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 4/7 of the square, which is 4/7 x 196 = 112 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 6 medium answer: 12 cm

The figure at right shows a rhombus overlapped with a square whose side length is 15 cm15\,\text{cm}. The area of the rhombus is 22 times the area of the overlapping region, and the area of the square is 55 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 15 cm15\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 15 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 15 cm square. The square is 5 times the overlap and the rhombus is 2 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 15 cm.
  • Square area = 5 x overlap area.
  • Rhombus area = 2 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 15 cm, so its area is side times side. Nothing else in the figure can be measured yet.
15×15=225 cm215 \times 15 = 225\ \text{cm}^2
225 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 5 times the overlap, so the overlap is the square's area divided by 5.
225÷5=45 cm2225 \div 5 = 45\ \text{cm}^2
The shared part is 45 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 2 times the overlap, so multiply the overlap by 2.
45×2=90 cm245 \times 2 = 90\ \text{cm}^2
The rhombus is 90 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=15 cmAC = 15\ \text{cm}
AC is 15 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 90 and AC = 15 gives 15 x BD / 2 = 90.
15×BD2=90  ⇒  15/2×BD=90  ⇒  BD=12 cm\dfrac{15 \times BD}{2} = 90 \;\Rightarrow\; 15/2 \times BD = 90 \;\Rightarrow\; BD = 12\ \text{cm}
BD is 12 cm.
Answer: 12 cm
4 · Reviewdoes it hold up?

Put 12 back into the formula: 15 x 12 / 2 = 90 cm2, and that is 2 times the 45 cm2 overlap, which is 225 / 5. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 2/5 of the square, which is 2/5 x 225 = 90 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 7 medium answer: 24 cm

The figure at right shows a rhombus overlapped with a square whose side length is 16 cm16\,\text{cm}. The area of the rhombus is 33 times the area of the overlapping region, and the area of the square is 44 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 16 cm16\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 16 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 16 cm square. The square is 4 times the overlap and the rhombus is 3 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 16 cm.
  • Square area = 4 x overlap area.
  • Rhombus area = 3 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 16 cm, so its area is side times side. Nothing else in the figure can be measured yet.
16×16=256 cm216 \times 16 = 256\ \text{cm}^2
256 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 4 times the overlap, so the overlap is the square's area divided by 4.
256÷4=64 cm2256 \div 4 = 64\ \text{cm}^2
The shared part is 64 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 3 times the overlap, so multiply the overlap by 3.
64×3=192 cm264 \times 3 = 192\ \text{cm}^2
The rhombus is 192 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=16 cmAC = 16\ \text{cm}
AC is 16 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 192 and AC = 16 gives 16 x BD / 2 = 192.
16×BD2=192  ⇒  8×BD=192  ⇒  BD=24 cm\dfrac{16 \times BD}{2} = 192 \;\Rightarrow\; 8 \times BD = 192 \;\Rightarrow\; BD = 24\ \text{cm}
BD is 24 cm.
Answer: 24 cm
4 · Reviewdoes it hold up?

Put 24 back into the formula: 16 x 24 / 2 = 192 cm2, and that is 3 times the 64 cm2 overlap, which is 256 / 4. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 3/4 of the square, which is 3/4 x 256 = 192 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 8 medium answer: 20 cm

The figure at right shows a rhombus overlapped with a square whose side length is 18 cm18\,\text{cm}. The area of the rhombus is 55 times the area of the overlapping region, and the area of the square is 99 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 18 cm18\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 18 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 18 cm square. The square is 9 times the overlap and the rhombus is 5 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 18 cm.
  • Square area = 9 x overlap area.
  • Rhombus area = 5 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 18 cm, so its area is side times side. Nothing else in the figure can be measured yet.
18×18=324 cm218 \times 18 = 324\ \text{cm}^2
324 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 9 times the overlap, so the overlap is the square's area divided by 9.
324÷9=36 cm2324 \div 9 = 36\ \text{cm}^2
The shared part is 36 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 5 times the overlap, so multiply the overlap by 5.
36×5=180 cm236 \times 5 = 180\ \text{cm}^2
The rhombus is 180 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=18 cmAC = 18\ \text{cm}
AC is 18 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 180 and AC = 18 gives 18 x BD / 2 = 180.
18×BD2=180  ⇒  9×BD=180  ⇒  BD=20 cm\dfrac{18 \times BD}{2} = 180 \;\Rightarrow\; 9 \times BD = 180 \;\Rightarrow\; BD = 20\ \text{cm}
BD is 20 cm.
Answer: 20 cm
4 · Reviewdoes it hold up?

Put 20 back into the formula: 18 x 20 / 2 = 180 cm2, and that is 5 times the 36 cm2 overlap, which is 324 / 9. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 5/9 of the square, which is 5/9 x 324 = 180 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 9 hard answer: 25 cm

The figure at right shows a rhombus overlapped with a square whose side length is 20 cm20\,\text{cm}. The area of the rhombus is 55 times the area of the overlapping region, and the area of the square is 88 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 20 cm20\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 20 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 20 cm square. The square is 8 times the overlap and the rhombus is 5 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 20 cm.
  • Square area = 8 x overlap area.
  • Rhombus area = 5 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 20 cm, so its area is side times side. Nothing else in the figure can be measured yet.
20×20=400 cm220 \times 20 = 400\ \text{cm}^2
400 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 8 times the overlap, so the overlap is the square's area divided by 8.
400÷8=50 cm2400 \div 8 = 50\ \text{cm}^2
The shared part is 50 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 5 times the overlap, so multiply the overlap by 5.
50×5=250 cm250 \times 5 = 250\ \text{cm}^2
The rhombus is 250 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=20 cmAC = 20\ \text{cm}
AC is 20 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 250 and AC = 20 gives 20 x BD / 2 = 250.
20×BD2=250  ⇒  10×BD=250  ⇒  BD=25 cm\dfrac{20 \times BD}{2} = 250 \;\Rightarrow\; 10 \times BD = 250 \;\Rightarrow\; BD = 25\ \text{cm}
BD is 25 cm.
Answer: 25 cm
4 · Reviewdoes it hold up?

Put 25 back into the formula: 20 x 25 / 2 = 250 cm2, and that is 5 times the 50 cm2 overlap, which is 400 / 8. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 5/8 of the square, which is 5/8 x 400 = 250 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 10 hard answer: 16 cm

The figure at right shows a rhombus overlapped with a square whose side length is 20 cm20\,\text{cm}. The area of the rhombus is 22 times the area of the overlapping region, and the area of the square is 55 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 20 cm20\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 20 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 20 cm square. The square is 5 times the overlap and the rhombus is 2 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 20 cm.
  • Square area = 5 x overlap area.
  • Rhombus area = 2 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 20 cm, so its area is side times side. Nothing else in the figure can be measured yet.
20×20=400 cm220 \times 20 = 400\ \text{cm}^2
400 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 5 times the overlap, so the overlap is the square's area divided by 5.
400÷5=80 cm2400 \div 5 = 80\ \text{cm}^2
The shared part is 80 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 2 times the overlap, so multiply the overlap by 2.
80×2=160 cm280 \times 2 = 160\ \text{cm}^2
The rhombus is 160 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=20 cmAC = 20\ \text{cm}
AC is 20 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 160 and AC = 20 gives 20 x BD / 2 = 160.
20×BD2=160  ⇒  10×BD=160  ⇒  BD=16 cm\dfrac{20 \times BD}{2} = 160 \;\Rightarrow\; 10 \times BD = 160 \;\Rightarrow\; BD = 16\ \text{cm}
BD is 16 cm.
Answer: 16 cm
4 · Reviewdoes it hold up?

Put 16 back into the formula: 20 x 16 / 2 = 160 cm2, and that is 2 times the 80 cm2 overlap, which is 400 / 5. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 2/5 of the square, which is 2/5 x 400 = 160 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 11 hard answer: 18 cm

The figure at right shows a rhombus overlapped with a square whose side length is 24 cm24\,\text{cm}. The area of the rhombus is 33 times the area of the overlapping region, and the area of the square is 88 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 24 cm24\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 24 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 24 cm square. The square is 8 times the overlap and the rhombus is 3 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 24 cm.
  • Square area = 8 x overlap area.
  • Rhombus area = 3 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 24 cm, so its area is side times side. Nothing else in the figure can be measured yet.
24×24=576 cm224 \times 24 = 576\ \text{cm}^2
576 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 8 times the overlap, so the overlap is the square's area divided by 8.
576÷8=72 cm2576 \div 8 = 72\ \text{cm}^2
The shared part is 72 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 3 times the overlap, so multiply the overlap by 3.
72×3=216 cm272 \times 3 = 216\ \text{cm}^2
The rhombus is 216 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=24 cmAC = 24\ \text{cm}
AC is 24 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 216 and AC = 24 gives 24 x BD / 2 = 216.
24×BD2=216  ⇒  12×BD=216  ⇒  BD=18 cm\dfrac{24 \times BD}{2} = 216 \;\Rightarrow\; 12 \times BD = 216 \;\Rightarrow\; BD = 18\ \text{cm}
BD is 18 cm.
Answer: 18 cm
4 · Reviewdoes it hold up?

Put 18 back into the formula: 24 x 18 / 2 = 216 cm2, and that is 3 times the 72 cm2 overlap, which is 576 / 8. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 3/8 of the square, which is 3/8 x 576 = 216 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.
Variant 12 hard answer: 24 cm

The figure at right shows a rhombus overlapped with a square whose side length is 30 cm30\,\text{cm}. The area of the rhombus is 44 times the area of the overlapping region, and the area of the square is 1010 times the area of the overlapping region. What is the length of segment BD, in cm\text{cm}?

Figure description: A square with side 30 cm30\,\text{cm} is on the right, and rhombus ABCD overlaps it on the left. The rhombus has its top vertex at A, left vertex at B, bottom vertex at C, and right vertex at D, where D lies inside the square. The shaded region is where the rhombus and the square overlap. Segment BD is the horizontal diagonal of the rhombus.

A B C D 30 cm
Show solution
1 · Understandwhat's really being asked

A rhombus overlaps a 30 cm square. The square is 10 times the overlap and the rhombus is 4 times it. We need the rhombus's horizontal diagonal BD.

Givens
  • The square's side is 30 cm.
  • Square area = 10 x overlap area.
  • Rhombus area = 4 x overlap area.
Unknowns
  • The length of the diagonal BD.
Constraints
  • All three areas are tied to one another through the overlap.
2 · Planchoose the strategy

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

Only one of the three areas can be worked out straight away. Get that one, step across to the others through the multiples given, and finish by running the rhombus formula backwards.

3 · Execute5 carry out the plan

1Find the area I already can: the square

#7 Identify Subproblems 4.MD.A.3
The square's side is 30 cm, so its area is side times side. Nothing else in the figure can be measured yet.
30×30=900 cm230 \times 30 = 900\ \text{cm}^2
900 cm2, and it is the only foothold.

2Step down to the overlap

#11 Work Backwards 4.OA.A.3
The square is 10 times the overlap, so the overlap is the square's area divided by 10.
900÷10=90 cm2900 \div 10 = 90\ \text{cm}^2
The shared part is 90 cm2.

3Step up to the rhombus

#7 Identify Subproblems 4.OA.A.3
The rhombus is 4 times the overlap, so multiply the overlap by 4.
90×4=360 cm290 \times 4 = 360\ \text{cm}^2
The rhombus is 360 cm2.

4Read one diagonal from the figure

#1 Draw a Diagram 6.G.A.1
In the picture the rhombus's vertical diagonal AC runs exactly along the left edge of the square, so AC has the same length as the square's side.
AC=30 cmAC = 30\ \text{cm}
AC is 30 cm, for free.

5Work backwards through the rhombus area formula to get BD

#11 Work Backwards 6.G.A.1
A rhombus's area is one diagonal times the other, divided by 2. Putting in area 360 and AC = 30 gives 30 x BD / 2 = 360.
30×BD2=360  ⇒  15×BD=360  ⇒  BD=24 cm\dfrac{30 \times BD}{2} = 360 \;\Rightarrow\; 15 \times BD = 360 \;\Rightarrow\; BD = 24\ \text{cm}
BD is 24 cm.
Answer: 24 cm
4 · Reviewdoes it hold up?

Put 24 back into the formula: 30 x 24 / 2 = 360 cm2, and that is 4 times the 90 cm2 overlap, which is 900 / 10. Every link holds.

Another way: The two multiples can be combined first: the rhombus is 4/10 of the square, which is 2/5 x 900 = 360 cm2 -- the same answer in one step instead of two.

Standardsmin grade 6
  • 4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The square's area, the one thing computable at the start.
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Stepping down to the overlap and back up to the rhombus.
  • 6.G.A.1 Find area of triangles, special quadrilaterals, and polygons by composing — Reading AC off the figure and unpicking the rhombus formula.
💡Takeaway. When nothing seems findable, find the one thing that is -- the rest usually steps off it.