← Cut a composite figure into rectangles · Area by Decomposition

Cut a composite figure into rectangles · 12 practice problems

3.MD.C.74.MD.A.35.NBT.B.7

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Generated variants — 12

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

Variant 1 easy

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 10.2 by 10.8 cm rectangle sitting on a wider one 8.4 cm tall. We need the total area.

Givens

The top rectangle is 10.2 cm wide and 10.8 cm tall.
The bottom one reaches 11.5 cm further left and 9.3 cm further right.
The bottom rectangle is 8.4 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 10.2 cm wide and 10.8 cm tall, so multiply the two side lengths.

10.2 \times 10.8 = 110.16 \text{ cm}^2

110.16 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 11.5 cm left of the top piece, then across the 10.2 cm shared middle, then 9.3 cm to the right. Add these three pieces.

11.5 + 10.2 + 9.3 = 31 \text{ cm}

The bottom is 31 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 31 cm wide and 8.4 cm tall, so multiply these decimal lengths.

31 \times 8.4 = 260.4 \text{ cm}^2

260.4 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

110.16 + 260.4 = 370.56 \text{ cm}^2

370.56 cm2 altogether.

Answer: 370.56 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 10 x 11 and the bottom roughly 31 x 8, which comes to about 358 cm2 -- close to 370.56, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 11.5, 10.2 and 9.3 wide, of heights 8.4, 19.2 and 8.4. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 2 easy

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 9.5 by 6 cm rectangle sitting on a wider one 12 cm tall. We need the total area.

Givens

The top rectangle is 9.5 cm wide and 6 cm tall.
The bottom one reaches 10.5 cm further left and 8.8 cm further right.
The bottom rectangle is 12 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 9.5 cm wide and 6 cm tall, so multiply the two side lengths.

9.5 \times 6 = 57 \text{ cm}^2

57 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 10.5 cm left of the top piece, then across the 9.5 cm shared middle, then 8.8 cm to the right. Add these three pieces.

10.5 + 9.5 + 8.8 = 28.8 \text{ cm}

The bottom is 28.8 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 28.8 cm wide and 12 cm tall, so multiply these decimal lengths.

28.8 \times 12 = 345.6 \text{ cm}^2

345.6 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

57 + 345.6 = 402.6 \text{ cm}^2

402.6 cm2 altogether.

Answer: 402.6 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 10 x 6 and the bottom roughly 29 x 12, which comes to about 408 cm2 -- close to 402.6, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 10.5, 9.5 and 8.8 wide, of heights 12, 18 and 12. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 3 easy

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 11.2 by 9.6 cm rectangle sitting on a wider one 10.8 cm tall. We need the total area.

Givens

The top rectangle is 11.2 cm wide and 9.6 cm tall.
The bottom one reaches 13.4 cm further left and 8.6 cm further right.
The bottom rectangle is 10.8 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 11.2 cm wide and 9.6 cm tall, so multiply the two side lengths.

11.2 \times 9.6 = 107.52 \text{ cm}^2

107.52 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 13.4 cm left of the top piece, then across the 11.2 cm shared middle, then 8.6 cm to the right. Add these three pieces.

13.4 + 11.2 + 8.6 = 33.2 \text{ cm}

The bottom is 33.2 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 33.2 cm wide and 10.8 cm tall, so multiply these decimal lengths.

33.2 \times 10.8 = 358.56 \text{ cm}^2

358.56 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

107.52 + 358.56 = 466.08 \text{ cm}^2

466.08 cm2 altogether.

Answer: 466.08 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 11 x 10 and the bottom roughly 33 x 11, which comes to about 473 cm2 -- close to 466.08, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 13.4, 11.2 and 8.6 wide, of heights 10.8, 20.4 and 10.8. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 4 easy

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 11.8 by 8 cm rectangle sitting on a wider one 13.5 cm tall. We need the total area.

Givens

The top rectangle is 11.8 cm wide and 8 cm tall.
The bottom one reaches 12.3 cm further left and 7.5 cm further right.
The bottom rectangle is 13.5 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 11.8 cm wide and 8 cm tall, so multiply the two side lengths.

11.8 \times 8 = 94.4 \text{ cm}^2

94.4 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 12.3 cm left of the top piece, then across the 11.8 cm shared middle, then 7.5 cm to the right. Add these three pieces.

12.3 + 11.8 + 7.5 = 31.6 \text{ cm}

The bottom is 31.6 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 31.6 cm wide and 13.5 cm tall, so multiply these decimal lengths.

31.6 \times 13.5 = 426.6 \text{ cm}^2

426.6 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

94.4 + 426.6 = 521 \text{ cm}^2

521 cm2 altogether.

Answer: 521 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 12 x 8 and the bottom roughly 32 x 14, which comes to about 544 cm2 -- close to 521, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 12.3, 11.8 and 7.5 wide, of heights 13.5, 21.5 and 13.5. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 5 medium

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 13.5 by 6.5 cm rectangle sitting on a wider one 11.8 cm tall. We need the total area.

Givens

The top rectangle is 13.5 cm wide and 6.5 cm tall.
The bottom one reaches 8.8 cm further left and 12.2 cm further right.
The bottom rectangle is 11.8 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 13.5 cm wide and 6.5 cm tall, so multiply the two side lengths.

13.5 \times 6.5 = 87.75 \text{ cm}^2

87.75 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 8.8 cm left of the top piece, then across the 13.5 cm shared middle, then 12.2 cm to the right. Add these three pieces.

8.8 + 13.5 + 12.2 = 34.5 \text{ cm}

The bottom is 34.5 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 34.5 cm wide and 11.8 cm tall, so multiply these decimal lengths.

34.5 \times 11.8 = 407.1 \text{ cm}^2

407.1 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

87.75 + 407.1 = 494.85 \text{ cm}^2

494.85 cm2 altogether.

Answer: 494.85 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 14 x 6 and the bottom roughly 34 x 12, which comes to about 492 cm2 -- close to 494.85, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 8.8, 13.5 and 12.2 wide, of heights 11.8, 18.3 and 11.8. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 6 medium

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 8.8 by 7.2 cm rectangle sitting on a wider one 12.5 cm tall. We need the total area.

Givens

The top rectangle is 8.8 cm wide and 7.2 cm tall.
The bottom one reaches 11.8 cm further left and 13.6 cm further right.
The bottom rectangle is 12.5 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 8.8 cm wide and 7.2 cm tall, so multiply the two side lengths.

8.8 \times 7.2 = 63.36 \text{ cm}^2

63.36 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 11.8 cm left of the top piece, then across the 8.8 cm shared middle, then 13.6 cm to the right. Add these three pieces.

11.8 + 8.8 + 13.6 = 34.2 \text{ cm}

The bottom is 34.2 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 34.2 cm wide and 12.5 cm tall, so multiply these decimal lengths.

34.2 \times 12.5 = 427.5 \text{ cm}^2

427.5 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

63.36 + 427.5 = 490.86 \text{ cm}^2

490.86 cm2 altogether.

Answer: 490.86 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 9 x 7 and the bottom roughly 34 x 12, which comes to about 471 cm2 -- close to 490.86, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 11.8, 8.8 and 13.6 wide, of heights 12.5, 19.7 and 12.5. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 7 medium

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 14 by 7.5 cm rectangle sitting on a wider one 10.5 cm tall. We need the total area.

Givens

The top rectangle is 14 cm wide and 7.5 cm tall.
The bottom one reaches 9.6 cm further left and 11.4 cm further right.
The bottom rectangle is 10.5 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 14 cm wide and 7.5 cm tall, so multiply the two side lengths.

14 \times 7.5 = 105 \text{ cm}^2

105 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 9.6 cm left of the top piece, then across the 14 cm shared middle, then 11.4 cm to the right. Add these three pieces.

9.6 + 14 + 11.4 = 35 \text{ cm}

The bottom is 35 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 35 cm wide and 10.5 cm tall, so multiply these decimal lengths.

35 \times 10.5 = 367.5 \text{ cm}^2

367.5 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

105 + 367.5 = 472.5 \text{ cm}^2

472.5 cm2 altogether.

Answer: 472.5 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 14 x 8 and the bottom roughly 35 x 10, which comes to about 462 cm2 -- close to 472.5, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 9.6, 14 and 11.4 wide, of heights 10.5, 18 and 10.5. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 8 medium

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 12.5 by 5.5 cm rectangle sitting on a wider one 9.6 cm tall. We need the total area.

Givens

The top rectangle is 12.5 cm wide and 5.5 cm tall.
The bottom one reaches 7.8 cm further left and 14.2 cm further right.
The bottom rectangle is 9.6 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 12.5 cm wide and 5.5 cm tall, so multiply the two side lengths.

12.5 \times 5.5 = 68.75 \text{ cm}^2

68.75 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 7.8 cm left of the top piece, then across the 12.5 cm shared middle, then 14.2 cm to the right. Add these three pieces.

7.8 + 12.5 + 14.2 = 34.5 \text{ cm}

The bottom is 34.5 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 34.5 cm wide and 9.6 cm tall, so multiply these decimal lengths.

34.5 \times 9.6 = 331.2 \text{ cm}^2

331.2 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

68.75 + 331.2 = 399.95 \text{ cm}^2

399.95 cm2 altogether.

Answer: 399.95 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 12 x 6 and the bottom roughly 34 x 10, which comes to about 412 cm2 -- close to 399.95, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 7.8, 12.5 and 14.2 wide, of heights 9.6, 15.1 and 9.6. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 9 hard

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 7.4 by 8.5 cm rectangle sitting on a wider one 13.2 cm tall. We need the total area.

Givens

The top rectangle is 7.4 cm wide and 8.5 cm tall.
The bottom one reaches 14.6 cm further left and 10.5 cm further right.
The bottom rectangle is 13.2 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 7.4 cm wide and 8.5 cm tall, so multiply the two side lengths.

7.4 \times 8.5 = 62.9 \text{ cm}^2

62.9 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 14.6 cm left of the top piece, then across the 7.4 cm shared middle, then 10.5 cm to the right. Add these three pieces.

14.6 + 7.4 + 10.5 = 32.5 \text{ cm}

The bottom is 32.5 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 32.5 cm wide and 13.2 cm tall, so multiply these decimal lengths.

32.5 \times 13.2 = 429 \text{ cm}^2

429 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

62.9 + 429 = 491.9 \text{ cm}^2

491.9 cm2 altogether.

Answer: 491.9 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 7 x 8 and the bottom roughly 32 x 13, which comes to about 472 cm2 -- close to 491.9, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 14.6, 7.4 and 10.5 wide, of heights 13.2, 21.7 and 13.2. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 10 hard

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 14.6 by 8.8 cm rectangle sitting on a wider one 9.2 cm tall. We need the total area.

Givens

The top rectangle is 14.6 cm wide and 8.8 cm tall.
The bottom one reaches 7.4 cm further left and 10.8 cm further right.
The bottom rectangle is 9.2 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 14.6 cm wide and 8.8 cm tall, so multiply the two side lengths.

14.6 \times 8.8 = 128.48 \text{ cm}^2

128.48 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 7.4 cm left of the top piece, then across the 14.6 cm shared middle, then 10.8 cm to the right. Add these three pieces.

7.4 + 14.6 + 10.8 = 32.8 \text{ cm}

The bottom is 32.8 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 32.8 cm wide and 9.2 cm tall, so multiply these decimal lengths.

32.8 \times 9.2 = 301.76 \text{ cm}^2

301.76 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

128.48 + 301.76 = 430.24 \text{ cm}^2

430.24 cm2 altogether.

Answer: 430.24 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 15 x 9 and the bottom roughly 33 x 9, which comes to about 432 cm2 -- close to 430.24, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 7.4, 14.6 and 10.8 wide, of heights 9.2, 18 and 9.2. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 11 hard

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 8.6 by 9.2 cm rectangle sitting on a wider one 15 cm tall. We need the total area.

Givens

The top rectangle is 8.6 cm wide and 9.2 cm tall.
The bottom one reaches 13.2 cm further left and 6.4 cm further right.
The bottom rectangle is 15 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 8.6 cm wide and 9.2 cm tall, so multiply the two side lengths.

8.6 \times 9.2 = 79.12 \text{ cm}^2

79.12 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 13.2 cm left of the top piece, then across the 8.6 cm shared middle, then 6.4 cm to the right. Add these three pieces.

13.2 + 8.6 + 6.4 = 28.2 \text{ cm}

The bottom is 28.2 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 28.2 cm wide and 15 cm tall, so multiply these decimal lengths.

28.2 \times 15 = 423 \text{ cm}^2

423 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

79.12 + 423 = 502.12 \text{ cm}^2

502.12 cm2 altogether.

Answer: 502.12 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 9 x 9 and the bottom roughly 28 x 15, which comes to about 501 cm2 -- close to 502.12, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 13.2, 8.6 and 6.4 wide, of heights 15, 24.2 and 15. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Variant 12 hard

Find the area, in $\text{cm}^2$ , of the figure shown at the right.

Your answer

Show solution

1 · Understandwhat's really being asked

A T-shaped figure is a 16 by 4.5 cm rectangle sitting on a wider one 14.5 cm tall. We need the total area.

Givens

The top rectangle is 16 cm wide and 4.5 cm tall.
The bottom one reaches 9.2 cm further left and 7.8 cm further right.
The bottom rectangle is 14.5 cm tall.

Unknowns

The area of the whole figure.

Constraints

The bottom rectangle's full width is not given -- only the two overhangs.

2 · Planchoose the strategy

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

Cut the T along the line where the narrow part meets the wide part. Each piece is then a plain rectangle, and the only real work is assembling the bottom one's width from three pieces.

3 · Execute5 carry out the plan

1Cut the figure into two rectangles

#1 Draw a Diagram 3.MD.C.7

Slice the figure along the line where the narrow top meets the wide bottom. That gives two shapes whose areas are just length times width, and no part of the figure belongs to both.

Two easy rectangles instead of one awkward T.

2Find the area of the top rectangle

#7 Identify Subproblems 5.NBT.B.7

The top rectangle is 16 cm wide and 4.5 cm tall, so multiply the two side lengths.

16 \times 4.5 = 72 \text{ cm}^2

72 cm2 up top.

3Find the full width of the bottom rectangle

#7 Identify Subproblems 4.MD.A.3

The bottom rectangle reaches 9.2 cm left of the top piece, then across the 16 cm shared middle, then 7.8 cm to the right. Add these three pieces.

9.2 + 16 + 7.8 = 33 \text{ cm}

The bottom is 33 cm across.

4Find the area of the bottom rectangle

#7 Identify Subproblems 5.NBT.B.7

The bottom rectangle is 33 cm wide and 14.5 cm tall, so multiply these decimal lengths.

33 \times 14.5 = 478.5 \text{ cm}^2

478.5 cm2 underneath.

5Add the two areas

#7 Identify Subproblems 5.NBT.B.7

The total area is the top rectangle's area plus the bottom rectangle's area.

72 + 478.5 = 550.5 \text{ cm}^2

550.5 cm2 altogether.

Answer: 550.5 cm²

4 · Reviewdoes it hold up?

Estimate it: the top is roughly 16 x 4 and the bottom roughly 33 x 14, which comes to about 526 cm2 -- close to 550.5, so no decimal point has slipped.

Another way: Cutting it vertically instead works too: three columns 9.2, 16 and 7.8 wide, of heights 14.5, 19 and 14.5. Same total, more pieces.

Standardsmin grade 5

3.MD.C.7 Relate area to the operations of multiplication and addition — Cutting the figure into non-overlapping rectangles and adding.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Assembling the bottom rectangle's width from three pieces.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Multiplying and adding the decimal lengths.

💡Takeaway. Cut an awkward shape into rectangles along a line that already exists in it, and every piece becomes length times width.

Where next?

← Back to the problem: Cut a composite figure into rectangles