AMC 10 · 2007 · #20

Grade 11 geometry-3d

Archetype Solid Volume by Decomposition └ Sub-archetype Fraction or Difference of a Bounding Solid

regular-octagonvolume-pyramidisosceles-right-trianglespatial-visualization spatial-visualizationconvert-to-algebra ↑ Prerequisites: regular-octagonvolume-pyramid

📏 Long solution 💡 4 insights

Problem

Every corner of a unit cube is sliced off so all six square faces become regular octagons. Find the combined volume of the pieces cut away.

Pick an answer.

(A)

$\frac{5\sqrt{2}-7}{3}$

(B)

$\frac{10-7\sqrt{2}}{3}$

(C)

$\frac{3-2\sqrt{2}}{3}$

(D)

$\frac{8\sqrt{2}-11}{3}$

(E)

$\frac{6-4\sqrt{2}}{3}$

How to solve

Strategy Visualize Spatial Relationships

The condition lives on the 2D faces but the question is about 3D volume, so Tool #17 (Visualize Spatial Relationships) is the entry point: one plane at one corner shows up as a single straight cut on each of the three squares meeting there, and that one cut is shared by two faces at once. Tool #4 (Introduce a Variable) names how deep the cut bites into each edge. The step this problem actually turns on is the one that is easiest to skip, because a picture makes it look like nothing: the statement never says the cuts are symmetric, and never says the eight corners are cut equally. Assume that and you have assumed most of the problem. So the plan is to let 'regular' do the forcing twice. Its equal-angles half pins each cut plane to a 45° lean, which makes it take the same bite out of all three edges at its corner; its equal-sides half then makes all eight corners share one depth and pins the common value. Only after that is the configuration unique, which is what makes 'the total volume' a legitimate thing to ask for, and running the argument backwards also shows such a slicing exists at all. Tool #13 (Convert to Algebra) turns the side-length equality into a one-line equation, and Tool #7 (Identify Subproblems) splits the volume into eight identical right-corner tetrahedra — after a check that they do not overlap.

1STEP 1

Name the cut, assume nothing

Name the cut without assuming anything about it.

V=(0,0,0), cut plane through (a,0,0),(0,b,0),(0,0,c): x/a+y/b+z/c=1

2STEP 2

The 135 degree angle forces a symmetric bite

The octagon's interior angle forces a symmetric bite.

interior = 180° - 360°/8 = 135° ⟹ cut triangle is 45°-45°-90° ⟹ a=b=c=t_V

3STEP 3

Equal sides force one common depth

Equal octagon sides give one equation for the depth.

slanted side = t√(2), straight side = 1-2t ⟹ 1-2t = t√(2)

4STEP 4

Solve for t and check it really works

Solving gives a depth that really works.

t=1/(2+√(2))=(2-√(2))/2=1-√(2)/2≈ 0.2929, octagon side=t√(2)=√(2)-1

5STEP 5

One corner piece is t cubed over six

One corner piece is a small pyramid.

V_corner=1/3Bh=1/3·t²/2 · t=t³/6

6STEP 6

Add eight pieces that cannot overlap

The eight pieces cannot overlap, so adding gives (10-7√(2))/3, choice (A).

8·t³/6=4t³/3=4/3·(10-7√(2))/4=(10-7√(2))/3≈ 0.0335

Answer

(10-7√(2))/3

Start with size. The total must lie strictly between 0 and 1, and since t ＜ 0.3 the eight pieces come to less than 8·0.3³/6=0.036. That single crude bound already kills (C) ≈ 0.0572, (D) ≈ 0.1046 and (E) ≈ 0.1144, leaving only (A) ≈ 0.0237 and (B) ≈ 0.0335 — so the last comparison genuinely needs the exact value (10-7√(2))/3=0.03350…, and it lands on (B). A second sanity check compares dimensions: each face loses four triangles of area t²/2, so 2t²=3-2√(2)≈ 0.172, about 17% of the face area, while only about 3.4% of the volume goes. That is exactly what should happen, because each corner piece fills only one sixth of the little t-box it sits in, so it is far thinner than its footprint suggests. The distractors are all near misses from this same computation, and they are worth naming. Choice (E) is 4t²/3 — the final formula 4t³/3 with the exponent dropped by one, that is, the answer you get if you forget you are in three dimensions. Choice (C) is 2t²/3, the same slip off by a further factor of 2. Choice (A) is ((√(2)-1)³)/3: it uses the octagon's side √(2)-1≈ 0.414 where the cut depth t≈ 0.293 belongs, and keeping those two lengths apart is the entire point of Steps 2 and 3. Finally, the shape of the answer is right: 10-7√(2) is a small positive number because 7√(2)=9.899…, so a solver who mis-signs the radical and writes (7√(2)-10)/3 gets a negative volume and knows immediately something broke.

💡Key takeaway

The word 'regular' does two separate jobs: equal angles force every corner cut to lean at 45° and bite all three edges equally, and equal sides then fix exactly how deep the bite is, after which the volume is just eight copies of one small corner.

Name the cut, assume nothing
The 135 degree angle forces a symmetric bite
Equal sides force one common depth
Solve for t and check it really works
One corner piece is t cubed over six
Add eight pieces that cannot overlap