Problem 24 · 2022 AMC 8
Hard
Geometry & Measurement
areaspatial-reasoning
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Answer: C — Volume 192.
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Hint 1 of 2
Don't get lost in the flat net — it folds into a triangular prism, whose volume is just (triangle base area) × (prism length). So you only need two things: the right-triangle's two legs and how long the prism is.
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Hint 2 of 2
Edges that fold to meet must be equal: AH = EF = 8 makes the rectangular faces 8 wide (the prism length). The triangle's legs are the prism length (8) and GJ = GH − HJ = 14 − 8 = 6.
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Approach: recognize the solid (triangular prism), then volume = base × length
- Insight: a net is a fold-up plan for a solid you already know — here a triangular prism. So skip the clutter and chase only two ingredients: the triangular base and the prism's length. Volume = (base area) × length.
- Edges that come together in the fold are equal. The rectangular faces are all 8 across (AB = BC = HJ = GF = 8, and BJ = AH = 8), so the prism length is 8.
- The triangular base is right triangle BJG. One leg is HJ = 8; the other is GJ = GH − HJ = 14 − 8 = 6. Base area = (6 × 8)/2 = 24.
- Volume = 24 × 8 = 192.
- You'll see this again: for any net problem, first name the 3-D shape it folds into, then track only the few lengths its volume formula needs — using “edges that meet are equal” to recover the ones not labeled.
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