Reference / Curriculum / 3B / Chapter 13

3B · Chapter 13 · Lesson 4

Area of Composite Figures

Decompose into rectangles or remove missing rectangles to find area.

MATH-3B-C13-L04Independent and tutor-supported

Today's mathematical question

A 4-by-5 rectangle has a 2-by-2 corner removed. Find the remaining area in square units.

Learning target

Decompose into rectangles or remove missing rectangles to find area.

Before starting: Prior lesson: Area of Rectangles. Try the warm-up independently.

Prerequisite · 3 questions

Check earlier skills first

These are earlier ideas you need for today; revisit them if they are difficult.

  1. P1

    A rectangle is 6 unit squares across and 5 unit squares down. What is its area?

    Reveal explanation

    Answer: 30 square units

    There are 5 rows of 6 equal square units: 6 × 5 = 30 square units.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  2. P2

    A rectangle is 9 unit squares across and 3 unit squares down. What is its area?

    Reveal explanation

    Answer: 27 square units

    There are 3 rows of 9 equal square units: 9 × 3 = 27 square units.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  3. P3

    A rectangle is 5 unit squares across and 5 unit squares down. What is its area?

    Reveal explanation

    Answer: 25 square units

    There are 5 rows of 5 equal square units: 5 × 5 = 25 square units.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.

Learn and understand

A composite figure can be broken into rectangles whose areas you already know. Split the shape into nonoverlapping rectangles and add their areas, or draw a surrounding large rectangle and subtract an empty corner. Both methods must count exactly the filled region. For an L-shaped region, start with a 6-by-5 enclosing rectangle and remove a 2-by-3 missing corner. The area is 30 − 6 = 24 square units.

Represent it three ways

  1. Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
  2. Pictorial: Mark the missing rectangular corner and calculate whole area minus removed area. Verify by counting visible grid squares.
  3. Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.

Important vocabulary

composite shape
figure built from smaller regions
decompose
break into simpler parts
area
space covered in square units

Fully worked examples and checks

Try each new understanding check before revealing the explanation.

Worked example 1

Explore, calculate, verify

A 4-by-5 rectangle has a 2-by-2 corner removed. Find the remaining area in square units.

Each visible square is 1 square unit.
Square-unit grid; count only visible squares without gaps or overlap.
  1. Identify the whole, the units, and what the question asks.
  2. Whole rectangle: 4 × 5 = 20. Missing corner: 2 × 2 = 4. Remaining: 20 − 4 = 16.
  3. Check that the answer uses the right units and makes sense.

Answer: 16 square units

Check your understanding 1

A 6-by-3 rectangle has a 2-by-1 corner removed. Find the remaining area in square units.

Reveal answer and explanation

16 square units — Whole rectangle: 6 × 3 = 18. Missing corner: 2 × 1 = 2. Remaining: 18 − 2 = 16.

Each visible square is 1 square unit.
Square-unit grid; count only visible squares without gaps or overlap.

Worked example 2

Explore, calculate, verify

A 4-by-4 rectangle has a 3-by-1 corner removed. Find the remaining area in square units.

Each visible square is 1 square unit.
Square-unit grid; count only visible squares without gaps or overlap.
  1. Identify the whole, the units, and what the question asks.
  2. Whole rectangle: 4 × 4 = 16. Missing corner: 3 × 1 = 3. Remaining: 16 − 3 = 13.
  3. Check that the answer uses the right units and makes sense.

Answer: 13 square units

Check your understanding 2

A 6-by-2 rectangle has a 2-by-1 corner removed. Find the remaining area in square units.

Reveal answer and explanation

10 square units — Whole rectangle: 6 × 2 = 12. Missing corner: 2 × 1 = 2. Remaining: 12 − 2 = 10.

Each visible square is 1 square unit.
Square-unit grid; count only visible squares without gaps or overlap.

Worked example 3

Explore, calculate, verify

A 8-by-5 rectangle has a 5-by-1 corner removed. Find the remaining area in square units.

Each visible square is 1 square unit.
Square-unit grid; count only visible squares without gaps or overlap.
  1. Identify the whole, the units, and what the question asks.
  2. Whole rectangle: 8 × 5 = 40. Missing corner: 5 × 1 = 5. Remaining: 40 − 5 = 35.
  3. Check that the answer uses the right units and makes sense.

Answer: 35 square units

Check your understanding 3

A 6-by-4 rectangle has a 4-by-3 corner removed. Find the remaining area in square units.

Reveal answer and explanation

12 square units — Whole rectangle: 6 × 4 = 24. Missing corner: 4 × 3 = 12. Remaining: 24 − 12 = 12.

Each visible square is 1 square unit.
Square-unit grid; count only visible squares without gaps or overlap.

Guided · 5 questions

Guided practice

Keep the model and earlier examples visible while practising.

  1. G1

    A 9-by-3 rectangle has a 2-by-1 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 25 square units

    Whole rectangle: 9 × 3 = 27. Missing corner: 2 × 1 = 2. Remaining: 27 − 2 = 25.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  2. G2

    A 9-by-3 rectangle has a 4-by-1 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 23 square units

    Whole rectangle: 9 × 3 = 27. Missing corner: 4 × 1 = 4. Remaining: 27 − 4 = 23.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  3. G3

    A 5-by-5 rectangle has a 1-by-4 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 21 square units

    Whole rectangle: 5 × 5 = 25. Missing corner: 1 × 4 = 4. Remaining: 25 − 4 = 21.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  4. G4

    A 9-by-5 rectangle has a 3-by-4 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 33 square units

    Whole rectangle: 9 × 5 = 45. Missing corner: 3 × 4 = 12. Remaining: 45 − 12 = 33.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  5. G5

    A 3-by-6 rectangle has a 2-by-3 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 12 square units

    Whole rectangle: 3 × 6 = 18. Missing corner: 2 × 3 = 6. Remaining: 18 − 6 = 12.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.

Independent · 8 questions

Independent practice

Work these without hints and explain the method used. Write answers on paper.

  1. I1

    A 9-by-4 rectangle has a 5-by-2 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 26 square units

    Whole rectangle: 9 × 4 = 36. Missing corner: 5 × 2 = 10. Remaining: 36 − 10 = 26.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  2. I2

    A 9-by-6 rectangle has a 7-by-2 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 40 square units

    Whole rectangle: 9 × 6 = 54. Missing corner: 7 × 2 = 14. Remaining: 54 − 14 = 40.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  3. I3

    A 3-by-6 rectangle has a 1-by-3 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 15 square units

    Whole rectangle: 3 × 6 = 18. Missing corner: 1 × 3 = 3. Remaining: 18 − 3 = 15.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  4. I4

    A 7-by-6 rectangle has a 1-by-1 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 41 square units

    Whole rectangle: 7 × 6 = 42. Missing corner: 1 × 1 = 1. Remaining: 42 − 1 = 41.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  5. I5

    A 6-by-4 rectangle has a 5-by-1 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 19 square units

    Whole rectangle: 6 × 4 = 24. Missing corner: 5 × 1 = 5. Remaining: 24 − 5 = 19.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  6. I6

    A 4-by-5 rectangle has a 3-by-3 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 11 square units

    Whole rectangle: 4 × 5 = 20. Missing corner: 3 × 3 = 9. Remaining: 20 − 9 = 11.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  7. I7

    A 5-by-3 rectangle has a 1-by-2 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 13 square units

    Whole rectangle: 5 × 3 = 15. Missing corner: 1 × 2 = 2. Remaining: 15 − 2 = 13.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  8. I8

    A 4-by-4 rectangle has a 1-by-3 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 13 square units

    Whole rectangle: 4 × 4 = 16. Missing corner: 1 × 3 = 3. Remaining: 16 − 3 = 13.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.

Challenge · 2 questions

Challenge and error analysis

Go beyond arithmetic: justify a claim, correct reasoning, or connect two representations.

  1. C1

    A 8-by-9 rectangle loses a 2-by-2 corner. Explain two ways to find the remaining area.

    Reveal explanation

    Answer: 68 square units

    Subtract the missing four squares from the bounding rectangle, or split the remaining shape into two nonoverlapping rectangles and add.

  2. C2

    Why must the missing corner of a composite rectangle be subtracted instead of included?

    Reveal explanation

    Answer: It is outside the figure

    Counting the full bounding rectangle would count squares not inside the figure.

Exit · 3 questions

Exit check

Solve independently, then return on another day to check retention.

  1. E1

    A 6-by-2 rectangle has a 4-by-1 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 8 square units

    Whole rectangle: 6 × 2 = 12. Missing corner: 4 × 1 = 4. Remaining: 12 − 4 = 8.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  2. E2

    A 9-by-4 rectangle has a 6-by-1 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 30 square units

    Whole rectangle: 9 × 4 = 36. Missing corner: 6 × 1 = 6. Remaining: 36 − 6 = 30.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
  3. E3

    A 6-by-3 rectangle has a 5-by-2 corner removed. Find the remaining area in square units.

    Reveal explanation

    Answer: 8 square units

    Whole rectangle: 6 × 3 = 18. Missing corner: 5 × 2 = 10. Remaining: 18 − 10 = 8.

    Each visible square is 1 square unit.
    Square-unit grid; count only visible squares without gaps or overlap.
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