Subtract a Removed Nonoverlapping Region
A composite region can be measured by taking a full enclosing rectangle and subtracting a removed square.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: A 6-by-4 rectangular region has one 2-by-2 square removed. Find the remaining area. Candidate answers: (A) 19 square units (B) 20 square units (C) 21 square units (D) 40 square units.
- Independent check inputs: A 9-by-6 rectangular region has one 3-by-3 square removed. Find the remaining area. Candidate answers: (A) 44 square units (B) 45 square units (C) 46 square units (D) 90 square units.
Learn the idea
Geometry can support probability, material, density and layout models when the assumptions are explicit. Ratios and decompositions must use nonoverlapping regions, consistent dimensions and the correct uniformity or exact-fit conditions. A composite region can be measured by taking a full enclosing rectangle and subtracting a removed square. The areas must use the same units and describe nonoverlapping pieces accurately.
Here is the method on the supplied teaching case. The full rectangle has area 6×4=24. Subtract the nonoverlapping removed square’s area 4: remaining area=20.
Subtract the removed area, not its perimeter. A drawing can help confirm that the square lies inside the region; overlapping removal shapes would require additional inclusion-and-exclusion reasoning.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: A 6-by-4 rectangular region has one 2-by-2 square removed. Find the remaining area. Candidate answers: (A) 19 square units (B) 20 square units (C) 21 square units (D) 40 square units. Solution: The full rectangle has area 6×4=24. Subtract the nonoverlapping removed square’s area 4: remaining area=20. The final answer is 20 square units. Subtract the removed area, not its perimeter. A drawing can help confirm that the square lies inside the region; overlapping removal shapes would require additional inclusion-and-exclusion reasoning.
Check your reasoning
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Helpful tips
- A composite region can be measured by taking a full enclosing rectangle and subtracting a removed square.
- Subtract the removed area, not its perimeter.
See the answer and explanation
45 square units
The full rectangle has area 9×6=54. Subtract the nonoverlapping removed square’s area 9: remaining area=45.
Take it further
Changed-input task: A 27-by-18 rectangular region has one 9-by-9 square removed. Find the remaining area. Candidate answers: (A) 404 square units (B) 405 square units (C) 406 square units (D) 810 square units. Show the mathematical reason, not only the selected answer. Expected result: 405 square units. The full rectangle has area 27×18=486. Subtract the nonoverlapping removed square’s area 81: remaining area=405.
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