Problem
Surface area of a composite solid
Two solids are drawn. Solid (1) is a cube with a small cube stuck on. Solid (2) is a block with a smaller block cut away. Find the surface area of each.
Reasoning · Grade 6-1 Surface Area and Volume of Solids
Your answer
Watch out A notch seems to shrink the surface, but the inner walls replace it exactly.
How to solve
Strategy Organize Information in More Ways — Adding up nine separate rectangles for solid (2) invites a mistake, and worse, the depth needed for those rectangles is not even printed on the figure. So instead of counting faces where they are, I move them: slide each inward-facing wall of the notch straight out until it fills the hole it left in the outside of the block. Every wall lands on a hole of exactly its own size, so the total area does not change at all - and what is left is a plain rectangular box whose surface area is one short formula. Solid (1) yields to the same idea in reverse, sliding the small cube's front face back into the front face of the big cube. Before any of that, the figure has to be read carefully (tools 17 and 1), because both answers depend on a reading the drawing only half states.
1STEP 1
Read the two figures before touching any arithmetic
Pin down each solid's lengths first.
depth = 3 + 2 = 5 cm, box = 9 × 5 × 8
A drawing of a solid always hides some of it, so the first job is to decide what the solid actually is. Every number after this depends on the 5 cm depth, which no label on the page states outright - it is the sum of two labels.
6.G.A.4Visualize Spatial Relationships2STEP 2
Solid (1): start with the plain cube
The plain cube in (1) is 54 cm².
6 × (3 × 3) = 6 × 9 = 54 cm²
Breaking a solid into the flat rectangles that make up its skin turns a three-dimensional question into a handful of easy length-times-width multiplications.
4.MD.A.3Identify Subproblems3STEP 3
Solid (1): what sticking the small cube on changes
The small cube hides 1 and adds 5.
54 - 1 + 5 = 58 cm²
Attaching a block always hides one square and reveals the block's other five, so the change is a small, countable swap rather than a whole new calculation.
7.G.B.6Identify Subproblems4STEP 4
Solid (1) again, by sliding a face - the method the rest of the problem needs
So (1) is 58 cm².
9 + (9 × 5) + (1 × 4) = 9 + 45 + 4 = 58 cm²
Sliding a face sideways never changes its area, so it is a free move - and the two ways of counting agreeing on 58 is a good sign the reading of the figure is sound.
6.G.A.4Organize Information In More WaysSliding each inner wall out to the surrounding box leaves the surface area exactly as it was.
Why?
Sliding a face keeps its size, so the area that moves out to the box is exactly the area that left the notch.
🧱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?
The whole surface is its faces put together, so trading one face for an equal one leaves the total untouched.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
5STEP 5
Solid (2): write down the notch
Write down the notch in (2).
removed block = 4 × 3 × 6, 8 - 2 = 6
Naming the notch's three measurements in the same order as the box's own three - width, depth, height - keeps each face pairing straight in the steps that follow.
6.G.A.4Draw A Diagram6STEP 6
Solid (2): slide the three inner walls out and watch nothing change
Sliding the three inner walls out leaves it unchanged.
4 × 3 = 12, 4 × 6 = 24, 3 × 6 = 18
The block was cut from a corner, which is what makes this work: each inner wall is parallel to an outer face and is exactly the size of the hole in it. Cut a notch out of the middle of a face instead and the extra side walls would have nowhere to go, and the surface area would grow.
7.G.B.6Organize Information In More Ways7STEP 7
Solid (2): the surface area of that box
That leaves the whole block's 314 cm².
2(9 × 5) + 2(9 × 8) + 2(5 × 8) = 90 + 144 + 80 = 314 cm²
Once the solid has been turned into an ordinary box, a single well-known formula finishes it - which is the whole reason for going to the trouble of moving the faces.
7.G.B.6Identify Subproblems8STEP 8
A note on the other reading of solid (1)
Sliding faces in (1) also gives 58 cm².
54 - 1 + 5 = 58 cm² either way
It is worth knowing when an ambiguity in a picture actually changes the answer and when it does not - here it does not, which is why the figure can get away with being terse.
7.G.B.6Visualize Spatial RelationshipsAnswer
58, 314 cm²
54 − 1 + 5 = 58
Check solid (2) the long way, face by face, without any sliding. Top: 9 x 5 = 45 minus the notch footprint 4 x 3 = 12, leaving 33. Notch floor: 12. Bottom: 45. Front: 9 x 8 = 72 minus the notch opening 4 x 6 = 24, leaving 48. Notch back wall: 24. Back: 72. Left: 5 x 8 = 40 minus the notch opening 3 x 6 = 18, leaving 22. Notch right wall: 18. Right: 40. The nine pieces total 33 + 12 + 45 + 48 + 24 + 72 + 22 + 18 + 40 = 314 square centimetres, matching. Solid (1) is sensible too: the plain cube is 54 and adding a small cube can only push the total up a little, and 58 - 54 = 4 is exactly the four side faces of a 1 cm cube. Both answers are in square centimetres, correct for area, and both are far larger than any single face (9 and 72 respectively) yet far smaller than any wild over-count, so the magnitudes sit where they should. Note also that the notch changes the volume of solid (2) a great deal - it removes 4 x 3 x 6 = 72 cubic centimetres out of 360 - while leaving the surface area untouched, which is exactly the point the problem is making.
Takeaway
Push the pushed-in faces back out: a bite taken from a corner leaves the surface area exactly what the whole box had.
- Read the two figures before touching any arithmetic
- Solid (1): start with the plain cube
- Solid (1): what sticking the small cube on changes
- Solid (1) again, by sliding a face - the method the rest of the problem needs
- Solid (2): write down the notch
- Solid (2): slide the three inner walls out and watch nothing change
- Solid (2): the surface area of that box
- A note on the other reading of solid (1)