Problem
Surface area of a complicated cube stack
Thirty-five cubes form a five-level stepped pyramid. Every level is pushed back into the same right-angle corner. The two rear walls are flat. Find the surface area of the solid.
Reasoning · Grade 6-2 Surface Area of Cube Stacks
Your answer
Watch out The steps look complicated, but from above they cover the same area as the base.
How to solve
Strategy Visualize Spatial Relationships — Counting 90 little squares one at a time off a drawing is exactly how this problem gets answered wrong -- faces get skipped behind steps or counted twice. The reliable route is to break the surface into six flat views and count squares in each: the shape is a staircase in every direction, so nothing hides, and each view is a small triangle or staircase whose squares are easy to count. Then, because face-counting can still go astray, the whole answer is re-derived a completely different way by counting the joints where two cubes touch, and the two answers are compared.
1STEP 1
Check the solid and set up the height of every column
Column heights confirm the 35 cubes.
15 + 10 + 6 + 3 + 1 = 35
Adding the five level sizes and getting 35 confirms the picture has been read correctly before any surface counting starts.
5.MD.C.4Identify Subproblems2STEP 2
Look from above and from below
From above 15 faces show.
5 + 4 + 3 + 2 + 1 = 15
The view from above is just the footprint, and its 15 squares can be counted as five rows of 5, 4, 3, 2 and 1.
3.MD.C.6Visualize Spatial Relationships3STEP 3
Look from the front and from the right side
The front and side each show 15 too.
5 + 4 + 3 + 2 + 1 = 15
A staircase never hides itself: every tread and every riser is visible from the front, so the outline you see and the faces you count are the same 15 squares.
3.MD.C.6Visualize Spatial Relationships4STEP 4
The three remaining directions match the first three
Opposite directions match, giving 90 faces.
(15 + 15 + 15) × 2 = 90
Pairing up opposite directions halves the work, and it is safe here precisely because the two rear walls are flat and every other face looks straight out.
6.G.A.4Visualize Spatial RelationshipsThe three remaining directions show exactly the same face counts as the first three, so only three views need counting.
Why?
Looking from the opposite side sees the same solid from behind, and no length or face is changed by walking round it.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Each column of the solid shows exactly one face in each of the two opposite directions, so the two counts must match.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
5STEP 5
Turn faces into area
Each face is 1, so the surface is 90.
90 × (1 × 1) = 90 in²
With 1-inch cubes the face count and the square-inch count happen to be the same number, but writing the multiplication keeps the answer labelled as an area rather than a count.
7.G.B.6Analyze The Units6STEP 6
Check by counting the joints instead of the faces
Counting joints also gives 90.
6 × 35 - 2 × 60 = 210 - 120 = 90
Counting what is hidden is often easier than counting what shows, and getting 90 a second way from completely different numbers is strong evidence that no face was skipped.
6.G.A.4Change Focus Count The ComplementAnswer
90 square inches
15 × 6 = 90
The answer is in square inches, the right unit for a surface area, and it is a whole number, which it must be since every face is exactly 1 square inch. The size is sensible: 35 separate loose cubes would show 210 square inches, and stacking them can only hide faces, so the answer had to be well under 210. It also has to beat the smallest possible box that could hold the solid -- a 5 by 5 by 5 cube has surface area 150 square inches and 125 cubes -- and 90 for a solid with only 35 cubes sits comfortably below that. The two independent counts agree exactly: 15 + 15 + 15 doubled from the six views, and 210 - 120 from the joints, both give 90 faces.
Takeaway
Count the squares in the top, front and side views, double the total because nothing on a staircase hides -- 15 + 15 + 15 doubled is 90 square inches.
- Check the solid and set up the height of every column
- Look from above and from below
- Look from the front and from the right side
- The three remaining directions match the first three
- Turn faces into area
- Check by counting the joints instead of the faces