Problem 25 · 2009 AMC 8
Hard
Geometry & Measurement
surface-area-viewsstaircase-decomposition
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is 12 foot from the top face. The second cut is 13 foot below the first cut, and the third cut is 117 foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
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Answer: E — 11 square feet.
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Hint 1 of 2
Those ugly cut fractions (1/2, 1/3, 1/17) are bait — the slabs together still came from one cube, so their heights always add to 1. Look from all 6 directions instead of tracking each slab's thickness.
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Hint 2 of 2
Add up the SILHOUETTE area seen from each face. Top and bottom each show four 1×1 squares; front and back each show a staircase whose four heights total 1 (a full unit); the two ends each show only the tallest step.
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Approach: project onto all 6 faces — heights conspire to 1
- Each piece keeps its 1×1 footprint. Top & bottom views: you see all four 1×1 squares — 4 sq ft from the top, 4 from the bottom.
- Front & back views: the staircase silhouette is four width-1 strips whose heights are the four slab thicknesses — and those came from slicing one unit cube, so they sum to exactly 1. Area = 1 each ⇒ 2 sq ft.
- Two end views: the slabs descend like a staircase, so from an end you see only the tallest piece A's cross-section, 1 × ½ = ½ — 1 sq ft for both ends.
- Total surface area = 4 + 4 + 2 + 1 = 11.
- Why this transfers: for a blocky solid, total surface area = sum of the six flat-on silhouette areas (no part of an axis-aligned surface ever hides behind another). And when pieces come from one whole, their dimensions secretly add back to that whole — so deliberately messy fractions often cancel.
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