Competition · AMC preparation · step 4 of 4

AMC 8 · 2008 · #16

Grade 6 geometry-3d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border
volume-rectangular-prismsurface-areaspatial-visualizationratio-proportion systematic-enumerationidentify-subproblems ↑ Prerequisites: volume-rectangular-prismsurface-area
📏 Short solution 💡 3 insights 📊 Diagram
Problem
Seven unit cubes are glued together: one central cube has six neighbors, one stuck to each of its six faces (above, below, left, right, front, back). Find the ratio of the total volume (cubic units) to the total surface area (square units).

Pick an answer.

(A)
:1 : 6
(B)
: 7 : 36
(C)
: 1 : 5
(D)
: 7 : 30
(E)
: 6 : 25

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Systematic Count

Tool #7 (Break into Subproblems) splits the ratio into two clean counts: volume first, then surface area. Volume is just the cube count. For surface area, Tool #13 (Systematic Count) handles each of the 7 cubes in turn, asking how many of its 6 faces are exposed. Tool #15 (Visualize / Use Symmetry) makes the count almost trivial: the center cube is hidden by all 6 neighbors, and by symmetry every outer cube is in the same situation — exactly 1 glued face and 5 exposed faces.

1STEP 1

Find the volume

Each of the 7 unit cubes has volume 1, so the total volume is 7 cubic units.

V = 7 × 1 = 7
2STEP 2

Count the outside faces

Apart, the 7 cubes show 7 × 6 = 42 faces; the 6 glued seams each hide 2 squares, leaving 30 exposed faces.

outside faces = 7 × 6 - 2 × 6 = 42 - 12 = 30
3STEP 3

Recount cube by cube

By symmetry the center cube shows 0 faces and each of the 6 outer cubes shows 5, giving 6 × 5 = 30 again.

S = 0 + 6 × 5 = 30
4STEP 4

Form the ratio

Volume to surface area is 7 to 30, and gcd(7, 30) = 1, so 730\frac{7}{30} is already in lowest terms → (D).

V : S = 7 : 30 → (D)
Answer
: 7 : 30
Sanity check with a single cube: volume 1, surface area 6, ratio 1:6 — that is choice (A). Adding a second cube doubles the volume to 2 but only adds 4 new exposed faces (6 + 6 - 2 = 10), so the ratio grows. Each new cube glued on adds 1 to the volume and 4 to the surface (gains 5 new faces, hides 1 old one). Starting from 1:6 and adding 6 cubes the same way gives volume 1 + 6 = 7 and surface area 6 + 6 × 4 = 30. The arithmetic matches, and (D) is consistent. Choices (A) 1:6 and (B) 7:36 correspond to forgetting that glued faces hide squares; (C) 1:5 matches a single cube minus one face, not this shape.
💡Key takeaway

Volume is just the cube count, and surface area is just face count — every glued seam hides exactly 2 unit squares. Spot that, and a 3D AMC 8 problem becomes a clean Grade 6 ratio.

  • Find the volume
  • Count the outside faces
  • Recount cube by cube
  • Form the ratio

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