AMC 10 · 2010 · #9

Grade 7 geometry-3d
Archetype Solid Volume by Decomposition └ Sub-archetype Fraction or Difference of a Bounding Solid
volume-rectangular-prismprinciple-of-inclusion-exclusionspatial-visualization complementary-counting ↑ Prerequisites: volume-rectangular-prism
📏 Medium solution 💡 3 insights
Problem
A cube has a square tunnel punched straight through the centre of each face. Find the volume left.

Pick an answer.

(A)
7
(B)
8
(C)
10
(D)
12
(E)
15
How to solve
Strategy Change Focus / Count the Complement

The leftover solid is an awkward frame shape, but the material that was removed is made of simple boxes. So instead of measuring what stays, measure what is taken away and subtract it from the full cube. The catch is that the three tunnels overlap, so counting the removed boxes is its own subproblem that needs care about the shared center.

1STEP 1

Volume of the full cube

The full cube is 27 cubic inches.

3 × 3 × 3 = 27
2STEP 2

Volume of one straight tunnel

One straight tunnel takes 12 away.

2 × 2 × 3 = 12
3STEP 3

Three tunnels share one center

All three tunnels share one central block.

2 × 2 × 2 = 8
4STEP 4

Add up the removed material correctly

Correcting the overlap removes 20 in total.

3 × 12 - 3 × 8 + 8 = 36 - 24 + 8 = 20
5STEP 5

Subtract to get what remains

Subtracting leaves 7, choice (A).

27 - 20 = 7
Answer
7
Build the leftover frame directly and check it matches. What survives is 8 corner cubes, each 0.5 by 0.5 by 0.5 (volume 0.125, total 1), plus 12 edge bars, each 2 by 0.5 by 0.5 (volume 0.5, total 6). Adding gives 1 + 6 = 7, the same as 27 minus 20. Also, 7 is the smallest choice, which fits a cube that has been hollowed out almost completely, leaving only a thin frame.
💡Key takeaway

When holes overlap, don't just add up what you drill out; count the shared part once so you don't erase too much.

  • Volume of the full cube
  • Volume of one straight tunnel
  • Three tunnels share one center
  • Add up the removed material correctly
  • Subtract to get what remains

More like this

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