Competition · AMC preparation · step 4 of 4

AMC 8 · 2025 · #8

Grade 8 geometry-3d
Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation
surface-areavolume-rectangular-prismpolyhedron-netsperfect-squares identify-subproblemsdimensional-analysis ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic
📏 Short solution 💡 3 insights 📊 Diagram
Problem
Isaiah cuts a cardboard cube open along some of its edges and flattens it into a single connected shape (a net). That flat shape has total area 18 square centimeters. Find the volume of the original cube in cubic centimeters.

Pick an answer.

(A)
$3\sqrt{3}$
(B)
6
(C)
9
(D)
$6\sqrt{3}$
(E)
$9\sqrt{3}$

AMC 8 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

The picture is a flat net, but the question asks about a 3D volume — Tool #17 (Visualize Spatial Relationships) is the bridge: mentally re-fold the net into the cube to see that the flat area is exactly the cube's surface area (no overlap, no gaps). From there Tool #7 (Identify Subproblems) splits the work into a clean chain: (a) total surface area → area of one face, (b) area of one face → side length s, (c) side length → volume V = s³. Tool #8 (Analyze the Units) keeps us honest as the units march from cm² (area) down to cm (length) and up to cm³ (volume).

1STEP 1

Fold the net into a cube

Re-folding the net covers the cube's outside exactly once, so the net's area equals the cube's surface area.

Surface Area = Area of net = 18 cm²
2STEP 2

Find one face's area

Divide the 18 cm² among the cube's 6 identical faces to get one face's area of 3 cm².

Area of one face = (18 cm²)/6 = 3 cm²
3STEP 3

Turn area into side length

Each face is a square, so take the square root of its area to recover the side: s = √(3) cm.

s² = 3 → s = √(3) cm
4STEP 4

Cube the side length

Cube the side with V = s³, pairing √(3)·√(3) = 3, so V = 3√(3) cm³ → (A).

V = s³ = (√(3))³ = √(3)·√(3)·√(3) = 3 · √(3) = 3√(3) cm³ → (A)
Answer
3√(3)
Quick estimate: √(3) ≈ 1.73, so s ≈ 1.73 cm. Then s² ≈ 3 cm² (matches one face) and s³ ≈ 1.73 × 3 = 5.2 cm³. Compare to 3√(3) ≈ 3 × 1.73 = 5.2 — same value, so the algebra and the estimate agree. The unit chain (cm² for face area, cm for side, cm³ for volume) is also consistent with what the question asks for.
💡Key takeaway

This AMC 8 problem only needs the Grade 8 square-root idea (√(3)·√(3)=3) you already know — the rest is just "net area = surface area" and a little division!

  • Fold the net into a cube
  • Find one face's area
  • Turn area into side length
  • Cube the side length

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