Area of Composite Figures
Semester 4B · Chapter 11 · Lesson 3 · MATH-4B-C11-L03
Before you begin: prerequisite check
Recall: 11.01 and 11.02. Attempt these questions independently first.
A rectangular garden has area 18 m² and width 3 m. Find its length.
Show answer and reasoning
Area = length×width. 18÷3=6 m.
A rectangular garden has area 132 m² and width 11 m. Find its length.
Show answer and reasoning
Area = length×width. 132÷11=12 m.
A rectangular garden has area 80 m² and width 8 m. Find its length.
Show answer and reasoning
Area = length×width. 80÷8=10 m.
Key vocabulary
- area
- region measured in square units
- perimeter
- distance around the outer boundary in linear units
- composite figure
- a shape assembled from simpler nonoverlapping regions
- dimension
- specified side length
Understand the concept
A composite rectilinear figure can be divided into rectangles that do not overlap. Its total area is the sum of those simpler areas. Another method surrounds it with one large rectangle and subtracts the missing portion. Suppose a 7-by-5 rectangle has a 2-by-3 corner removed: the area is 35−6=29 square units. Label each side and be certain that every interior square is counted once. Two different decompositions should agree; checking both is good mathematical reasoning.
Fully worked examples and immediate checks
Read each example, predict how to begin, then follow its numerical explanation. The check below it is a fresh problem.
Worked example 1
An outer 9-by-8 rectangle has a 6-by-2 rectangular corner removed. Find the remaining area in square units.
Outer area 9×8=72; missing area 6×2=12. Subtract to get 60.
Check 1: try it yourself
An outer 12-by-8 rectangle has a 8-by-4 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 12×8=96; missing area 8×4=32. Subtract to get 64.
Worked example 2
An outer 12-by-9 rectangle has a 10-by-3 rectangular corner removed. Find the remaining area in square units.
Outer area 12×9=108; missing area 10×3=30. Subtract to get 78.
Check 2: try it yourself
An outer 9-by-8 rectangle has a 7-by-2 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 9×8=72; missing area 7×2=14. Subtract to get 58.
Worked example 3
An outer 7-by-6 rectangle has a 4-by-2 rectangular corner removed. Find the remaining area in square units.
Outer area 7×6=42; missing area 4×2=8. Subtract to get 34.
Check 3: try it yourself
An outer 7-by-9 rectangle has a 4-by-7 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 7×9=63; missing area 4×7=28. Subtract to get 35.
Guided practice
An outer 11-by-9 rectangle has a 4-by-6 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 11×9=99; missing area 4×6=24. Subtract to get 75.
An outer 11-by-6 rectangle has a 9-by-4 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 11×6=66; missing area 9×4=36. Subtract to get 30.
An outer 11-by-5 rectangle has a 6-by-3 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 11×5=55; missing area 6×3=18. Subtract to get 37.
An outer 10-by-7 rectangle has a 8-by-4 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 10×7=70; missing area 8×4=32. Subtract to get 38.
An outer 9-by-9 rectangle has a 7-by-5 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 9×9=81; missing area 7×5=35. Subtract to get 46.
Independent practice
An outer 7-by-5 rectangle has a 5-by-3 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 7×5=35; missing area 5×3=15. Subtract to get 20.
An outer 7-by-6 rectangle has a 5-by-4 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 7×6=42; missing area 5×4=20. Subtract to get 22.
An outer 8-by-8 rectangle has a 3-by-3 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 8×8=64; missing area 3×3=9. Subtract to get 55.
An outer 8-by-8 rectangle has a 2-by-2 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 8×8=64; missing area 2×2=4. Subtract to get 60.
An outer 8-by-7 rectangle has a 5-by-3 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 8×7=56; missing area 5×3=15. Subtract to get 41.
An outer 9-by-6 rectangle has a 5-by-2 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 9×6=54; missing area 5×2=10. Subtract to get 44.
An outer 9-by-5 rectangle has a 7-by-2 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 9×5=45; missing area 7×2=14. Subtract to get 31.
An outer 8-by-9 rectangle has a 5-by-5 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 8×9=72; missing area 5×5=25. Subtract to get 47.
Challenge and transfer
Use a diagram or verbal justification where appropriate. The goal is transfer rather than speed.
An outer 7-by-6 rectangle has a 5-by-3 rectangular corner removed. Find the remaining area in square units. Explain why your units and method are appropriate.
Show answer and reasoning
Outer area 7×6=42; missing area 5×3=15. Subtract to get 27. For area count square units; for perimeter trace only the outside boundary.
A learner claims: “For an 8-by-6 rectangle missing a 2-by-3 corner, the area is still 48 square units.” Identify what is wrong or incomplete and give a mathematically correct explanation.
Show answer and reasoning
False: subtract the missing 6 square units to get 42 square units. Explain the relevant quantities or properties, not merely the final answer.
Exit check
An outer 11-by-6 rectangle has a 3-by-3 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 11×6=66; missing area 3×3=9. Subtract to get 57.
An outer 9-by-8 rectangle has a 6-by-3 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 9×8=72; missing area 6×3=18. Subtract to get 54.
An outer 7-by-7 rectangle has a 5-by-5 rectangular corner removed. Find the remaining area in square units.
Show answer and reasoning
Outer area 7×7=49; missing area 5×5=25. Subtract to get 24.
Mastery check: Solve a new instance without hints, explain why the method works, and return later to confirm retention.