AMC 10 · 2015 · #17

Grade 7 geometry-3d
Archetype Solid Volume by Decomposition └ Sub-archetype Sum of Primitive Solids
spatial-visualizationvolume-rectangular-prismarea-triangles identify-subproblems ↑ Prerequisites: volume-rectangular-prism
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A box (right rectangular prism) has edge lengths 3, 4, and 5. Mark the center point of each of its six faces, then connect those six centers to build an octahedron. Find the volume of that octahedron.

Pick an answer.

(A)
$\dfrac{75}{12}$
(B)
10
(C)
12
(D)
$10\sqrt2$
(E)
15

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

This is a 3D shape hiding inside another 3D shape, so the load-bearing move is tool #17 (Visualize Spatial Relationships): picture where the six face centers actually sit. To pin that mental picture down with numbers, tool #1 (Draw a Diagram) puts the box in coordinates and reads off the centers. Once the picture is clear, the octahedron splits cleanly at its middle into two identical pyramids that share one flat base — tool #7 (Identify Subproblems): find the area of that shared base and the height of one pyramid, compute one pyramid's volume, and double it.

1STEP 1

Put the box in coordinates

Set the box from (0,0,0) to (5,4,3); each face center is that face's midpoint, so every coordinate is an end value or a halfway value.

(0,2,1.5), (5,2,1.5), (2.5,0,1.5), (2.5,4,1.5), (2.5,2,0), (2.5,2,3)
2STEP 2

Spot the two-pyramid shape

Four centers share height z=1.5 and frame a flat middle base; the other two sit right below and above it — two pyramids glued base-to-base.

3STEP 3

Find the shared base area

The middle base is a rhombus whose perpendicular diagonals measure 5 and 4, so its area is half their product, 10.

base area = 1/2 d₁ d₂ = 1/2· 5 · 4 = 10
4STEP 4

Find one pyramid's height and finish

Each pyramid rises 1.5 from the base to an apex at z=0 or z=3, so one is 1/3·10·1.5 = 5, and the octahedron doubles it to 10 — choice (B).

V = 2·1/3 (10)(1.5) = 2· 5 = 10
Answer
10
Compare against the whole box, whose volume is 3·4·5 = 60. The octahedron came out to 10, which is exactly one sixth of the box. That fraction is not an accident: for any box, joining face centers always yields an octahedron of one sixth the box volume, because the rhombus base is half the area of a cross-section and the two pyramids together use the third dimension at the one-third pyramid factor (1/2·1/3· 2 = 1/6). Getting a clean 10, well below the box's 60 and matching this known sixth-ratio, confirms the result.
💡Key takeaway

Cut a tricky 3D shape across its middle: this octahedron is just two equal pyramids sharing a flat base, so find the base area, the height, and double one pyramid.

  • Put the box in coordinates
  • Spot the two-pyramid shape
  • Find the shared base area
  • Find one pyramid's height and finish

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