AMC 8 · 2012 · #21

Grade 6 geometry-3d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

surface-areaarea-rectanglesspatial-visualization identify-subproblemsarea-difference ↑ Prerequisites: area-rectanglessurface-area

📏 Short solution 💡 2 insights

Problem

A white cube has edge length 10 feet. Marla paints a green border around a white square centered on every face, using up all of her green paint, which is enough to cover exactly 300 square feet. What is the area of one white square, in square feet?

Pick an answer.

(A)

$5\sqrt{2}$

(B)

(C)

$10\sqrt{2}$

(D)

(E)

$50\sqrt{2}$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Use Symmetry

Every face of the cube is treated the same way, so by symmetry the 300 sq ft of green paint splits evenly into 6 equal portions — one per face. That is Tool #11 (Use Symmetry): instead of tracking the whole cube, we only have to think about one face. Then Tool #7 (Identify Subproblems) handles that one face: its 100 sq ft area is made of two pieces, white square + green border, so the white area is just 100 minus the green portion. No square roots or Pythagoras needed — the radical choices are distractors.

1STEP 1

Each face of the cube is a 10-by-10 square, so one face has area 100 sq ft.

face area = 10 × 10 = 100 sq ft

2STEP 2

By symmetry the 300 sq ft of green paint splits evenly over 6 faces, so each face gets 50 sq ft of green.

green per face =

\frac{300}{6}

= 50 sq ft

3STEP 3

On one face the centered white square and green border fill it with no gaps, so their areas sum to 100 sq ft.

white area + green area = 100

4STEP 4

Subtract the green from the whole face: white area = 100 - 50 = 50 sq ft.

white area = 100 - 50 = 50 sq ft → (D)

Answer

Total surface area of the cube is 6 × 100 = 600 sq ft, and Marla's 300 sq ft of green paint is exactly half of that. So on every face, green covers half (50 sq ft) and white covers the other half (50 sq ft). That matches 50 in answer (D), and the white square is smaller than the full face (100 sq ft) but takes up a meaningful chunk — a believable "centered square with border" picture. The radical answers 5√(2), 10√(2), 50√(2) are traps for students who guess the white square is tilted 45° inside the face; nothing in the problem forces a tilt.

💡Key takeaway

This AMC 8 problem only needs the Grade 6 idea that a cube has 6 equal faces — then it's just 100 - 50 = 50 on one face.

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