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Case studies ／ Arithmetic

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The volume of a triangular pyramid from its net

Fold a 12 cm square into a triangular pyramid and find its volume. Solved in three steps, starting from spotting which edges become perpendicular when folded.

　Domain: Arithmetic

Analysis data (cue → response texts, stage analyses, explanation steps) is shown in the original Japanese. English rendering of the data layer is pending; titles, summaries and observations are in English.

PROBLEM

The problem

SubjectThe figure on the right is a square with sides of 12 cm, which is the net of a triangular pyramid. Points E and F are the midpoints of sides AB and AD. Find the volume of the pyramid in cm³. ( cm³)

CREATE BASE arithmetic materials (solid geometry, Japanese junior-high entrance level). The original figure is omitted; redrawn figures used for the analysis are shown.

OUTPUT

Judgment flowchart

Boxes are information; gold bands are judgments (cue → response). Starting from the givens, judgments build up to the answer.

ProblemThe figure shows a square with sides of 12 cm, which is the net of a triangular pyramid. Points E and F are the midpoints of sides AB and AD. Find the volume of the pyramid in cm³.

Square side 12 cmE, F: midpoints of AB, ADFolding a net → look for right anglesFolded, A, B and D meet at P;PE, PF, PC are mutually perpendicular (6, 6, 12 cm)Triangle area → base × height ÷ 2Base PEF area 18 cm²Pyramid volume → base area × height ÷ 3Volume 72 cm³

GivenDerived informationAnswerChain (cue → response)