Reasoning · Grade 6-1 Surface Area and Volume of Solids
Problem
Apart, the four cubes give 24.
Because every face is one square inch, area never needs a formula here — counting squares is counting area, so 'surface area' and 'number of exposed faces' are the same number.
3.MD.C.6Solve An Easier Related ProblemOne join hides 2.
The surface of a solid is just the collection of faces that touch the air. Joining hides one face from each of the two cubes, so it always removes the same two square inches, which is what makes the rule so easy to use.
6.G.A.4Solve An Easier Related ProblemEvery join between two cubes hides two square faces, so each join costs the surface exactly two squares.
A joined face belongs to two cubes at once and is counted once from each, so one join removes one face from each of them.
The surface is all the separate cubes' faces put together minus the ones the joins have hidden.
So the area is 24 minus twice the joins.
This turns a 3-D area question into simple counting, and it also says something worth noticing: the more joins a shape has, the smaller its surface area.
7.G.B.6Solve An Easier Related ProblemFive shapes have 3 joins.
Five of these shapes are chains or branches — you can walk from cube to cube and never come back to where you started, and such a shape of 4 cubes always has exactly 3 joins. Only the 2 by 2 square closes into a loop, and the loop gives it a fourth join.
7.G.B.6Make A Systematic ListOnly E has 4 joins.
Once the joins are counted, the rest is one subtraction per shape, and lining the six answers up in a row makes the odd one impossible to miss.
4.OA.A.3Make A Systematic ListSo E's surface is 2 smaller.
Packing cubes closer together — closer to a compact block — always tucks more faces out of sight, and the 2 by 2 square is the most compact of these six.
7.G.B.6Visualize Spatial RelationshipsEvery place two cubes touch hides 2 square inches, so the shape with the most joins - the 2 by 2 block - has the smallest surface.