A student calculates the area of an L-shaped figure. The bounding rectangle (the smallest rectangle that surrounds the L) would be 10 × 8 = 80 square units. The rectangular notch cut from the corner is 4 × 3 = 12 square units. The student writes 80 + 12 = 92 square units as the answer. What went wrong?
AThe student should have multiplied both areas together instead
BThe student added the notch area instead of subtracting it — the notch is the part that is missing, so it must be removed from the total
CThe subtraction method does not work for L-shapes; only splitting into rectangles is valid
DThe student calculated the bounding rectangle incorrectly
The subtraction strategy works by finding the area of the large containing rectangle, then removing the area of the missing piece. The notch is not part of the shape — it is a chunk that was cut away — so its area must be subtracted, not added. 80 − 12 = 68 square units is the correct answer. Adding the notch is a common error that results from misunderstanding what the notch represents: a missing piece, not an extra piece.
Question 2 Multiple Choice
When decomposing an L-shape into two rectangles to find total area, a student gets the wrong answer even though her multiplication is correct. What is the most likely cause?
AShe should have used addition instead of multiplication for each rectangle's area
BShe forgot to add the two rectangle areas at the end
CShe used incorrect side lengths for the component rectangles, likely because she did not figure out the missing dimensions first
DL-shapes cannot be decomposed into exactly two rectangles
Correct multiplication is only useful if the dimensions fed into it are correct. The critical prerequisite step is determining any missing side lengths from the labeled dimensions. In rectilinear shapes, opposite sides along any straight line must total the same amount, which is how missing lengths are derived. Skipping this step means the rectangle dimensions are wrong, and even perfect multiplication produces the wrong area.
Question 3 True / False
To find the area of a rectilinear shape, you should typically split it into rectangles and add the parts — the subtraction method is not a valid alternative.
TTrue
FFalse
Answer: False
Both strategies are equally valid and always produce the same answer. The 'add the parts' method decomposes the shape into non-overlapping rectangles, computes each area, and sums them. The 'subtract the missing piece' method computes the area of a bounding rectangle and subtracts the rectangular notch. The choice between them depends on which side lengths are easier to work with — not on which method is more correct.
Question 4 True / False
In a rectilinear shape, you can calculate any missing side length by looking at the dimensions on the opposite side of the figure.
TTrue
FFalse
Answer: True
Because all angles in a rectilinear shape are right angles, every horizontal span must be accounted for by horizontal dimensions, and every vertical span by vertical dimensions. Opposite sides along a straight run must sum to the same total. This means any unlabeled side length can be found by subtracting the labeled partial lengths on the same side from the total length given on the opposite side.
Question 5 Short Answer
Why is finding the missing side lengths the most critical step when calculating the area of an L-shaped figure?
Think about your answer, then reveal below.
Model answer: The missing side lengths define the dimensions of the component rectangles. Without them, you cannot compute any rectangle's area — you do not know what to multiply. In an L-shape, not all sides are labeled; you must derive the unlabeled lengths from the labeled ones by recognizing that opposite sides must account for the same total span. If you skip this step, every area calculation that follows will be based on wrong dimensions, no matter how accurately you apply length × width.
This step is where most errors in rectilinear area problems originate. The arithmetic (multiplication and addition or subtraction) is usually straightforward; the conceptual challenge is seeing that the shape's right-angle structure provides enough information to determine all missing lengths before any calculation begins.