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Counting the same volume twice

A total height or a missing notch can make a solid look larger than it is. Label each separate prism before multiplying.

Spot the wrong turn

Two nonoverlapping prisms measure 4 × 3 × 2 cm and 2 × 3 × 5 cm. Find their total volume.

Incorrect answer: 84 cubic cm

Do not use the total height as the upper prism’s height. That counts part of the base twice. Shared faces do not add volume, and the answer uses cubic units, not square units.

Work it through

The correction

54 cubic cm

The volumes are 24 and 30 cubic cm; 24 + 30 = 54.

Mark a split that leaves two rectangular prisms. Each part needs three dimensions.
Find missing lengths by subtracting aligned lengths from a total; do not guess from the drawing.
Multiply within each prism, add the results, and label the answer in cubic units.

Try the reasoning in another example

A solid fills a 10 × 4 × 6 cm box except for a 3 × 4 × 2 cm rectangular notch. Find its volume.

The full box would hold 10 × 4 × 6 = 240 cubic cm.
The empty notch is 3 × 4 × 2 = 24 cubic cm.
Subtract the empty part: 240 − 24.

Answer: 216 cubic cm

Explain why each step preserves the quantity or relationship. If you can compute the answer but cannot explain the step, revisit the model before adding more questions.

The short check below reuses the first example as a retrieval question and adds a second question. It checks this method; it does not establish a grade level.

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