Problem
Same boxes, different stacking, different surface area
Identical cube-shaped boxes are stacked into two different prisms. We need which has the larger surface area and by how much.
Geometry
Your answer
How to solve
Strategy Draw a Diagram — Counting boxes will not answer this -- surface area depends on the shape, not the count. Turn each stack into its three edge lengths in centimeters and use the same formula on both.
1STEP 1
Turn each stack into edge lengths
Each box adds 2 cm in the direction it is stacked, so A is a 4 cm cube and B is a tall 2 by 2 by 8 column.
2 × 2 = 4, 2 × 4 = 8
A is 4 × 4 × 4; B is 2 × 2 × 8.
6.G.A.2Visualize Spatial Relationships2STEP 2
Find A's surface area
A cube has six identical square faces.
4 × 4 × 6 = 96
A has 96 cm² of surface.
6.G.A.4Draw A Diagram3STEP 3
Find B's surface area
The column has two small square ends and four long side faces.
2 × 2 × 2 = 8, 2 × 8 × 4 = 64
B has 8 + 64 = 72 cm² of surface.
6.G.A.4Draw A Diagram4STEP 4
Compare them
Subtract the smaller surface area from the larger.
96 - 72 = 24
A is larger by 24 cm².
6.G.A.4Identify SubproblemsMore boxes does not mean more surface.
Why?
Surface area counts only the faces on the outside, and where two boxes meet both of their touching faces stop being outside.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Stacking eight boxes into a block hides more faces than standing four in a line does, so the bigger pile can still show less surface.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Answer
A, by 24 cm²
Count faces instead: A shows 24 box faces on the outside and B shows 18, and each box face is 4 cm² -- 96 against 72.
Takeaway
Surface area depends on shape, not on how many blocks went in.
- Turn each stack into edge lengths
- Find A's surface area
- Find B's surface area
- Compare them