Problem 16 · 2008 AMC 8
Medium
Geometry & Measurement
volume-surface-3d
A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?
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Answer: D — 7 : 30.
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Hint 1 of 2
See the shape as one center cube with a cube glued to each of its 6 faces — the center is buried with nothing showing.
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Hint 2 of 2
Don't count all 42 faces and then subtract; count what's actually exposed: each outer cube hides exactly the one face that touches the center.
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Approach: count only the exposed faces
- Volume is easy: 7 unit cubes = 7. For surface area, notice each of the 6 outer cubes touches the center on one face, so it shows 6 − 1 = 5 faces; the center cube shows nothing.
- Exposed faces: 6 × 5 = 30, and the center adds 0. So volume : surface = 7 : 30.
- Why this transfers: for glued-cube surface area, every shared face hides 2 unit squares (one on each cube) — counting exposed faces directly is faster than total-minus-hidden.
Another way — total faces minus hidden:
- 7 cubes have 7 × 6 = 42 faces. There are 6 glued joints, each hiding 2 faces, so 12 faces vanish.
- Exposed: 42 − 12 = 30, giving the ratio 7 : 30.
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