A cube of side 5 cm is painted on all its faces. If it is sliced into 1 cm3 cubes, how many of these 1 cm3 cubes have
- (i) exactly three faces painted?
- (ii) exactly two faces painted?
- (iii) exactly one face painted?
- (iv) no face painted?
Step-by-step solution
Idea: There are 5 × 5 × 5 = 125 small cubes. A small cube’s painted faces depend on where it sits: corner (3), along an edge but not a corner (2), in the middle of a face (1), or hidden inside (0). Each edge has 5 − 2 = 3 non-corner cubes; each face has a 3 × 3 middle.
(i) exactly three faces painted?
- Only corner cubes touch three outside faces. A cube has 8 corners, so 8 small cubes.1 mark
(ii) exactly two faces painted?
- These lie along an edge but not at its ends. Each edge has 5 − 2 = 3 such cubes, and a cube has 12 edges: 12 × 3 = 36.1 mark
(iii) exactly one face painted?
- These lie in the middle of a face, away from its edges: a (5 − 2) × (5 − 2) = 3 × 3 = 9 square on each face. 6 faces: 6 × 9 = 54.1 mark
(iv) no face painted?
- These are hidden inside, a cube of side 5 − 2 = 3: 33 = 27. Check: 8 + 36 + 54 + 27 = 125 = 53 ✓.1 mark
Check: 8 + 36 + 54 + 27 = 125 = 5 × 5 × 5, the total number of small cubes ✓.
Answer to write in the exam
(i)
Three faces painted: corner cubes
∴ Number = 8
(ii)
Two faces painted: edge cubes (not corners)
Each edge: 5 − 2 = 3
∴ Number = 12 × 3 = 36
(iii)
One face painted: middle of each face
Each face: (5 − 2)2 = 9
∴ Number = 6 × 9 = 54
(iv)
No face painted: inner cube of side 5 − 2 = 3
∴ Number = 33 = 27
Common mistakes that cost marks
- Counting 5 cubes per edge for two painted faces, which counts the corners again. Remove the 2 corners: 3 per edge.
- Using 4 edges or 6 edges instead of the cube’s 12 edges.
- For no face painted, taking the inner cube as side 4 (5 − 1) instead of 3: one layer is removed from each side.
How this can come in the exam
A cube of side 4 cm is painted on all faces and cut into 1 cm cubes. How many small cubes have no face painted?
- 8
- 16
- 27
- 24
Show answer
(A) 8
Inner cube of side 4 − 2 = 2: 23 = 8.
A painted cube of side 6 cm is cut into 1 cm cubes. How many small cubes have exactly two faces painted, and how many exactly one?
Show answer
Two faces: 12 × (6 − 2) = 48. One face: 6 × (6 − 2)2 = 6 × 16 = 96.Try one yourself
A cube of side n cm is painted and cut into 1 cm cubes. Write the number with 0, 1, 2 and 3 faces painted.
Show answer
0 faces: (n − 2)3; 1 face: 6(n − 2)2; 2 faces: 12(n − 2); 3 faces: 8. (For n = 5: 27, 54, 36, 8.)
More questions like this
- Find a cuboid with edges whose lengths are integers (in cm), given that it has a total surface area of exactly 100 cm2.
- Suppose a swimming pool is being built in your school. Two choices are available for the dimensions of the pool: (A) 8 ft (depth) × 50 ft × 20 ft and (B) 6 ft (depth) × 100 ft × 15 ft. Which option would be suitable/appropriate for your school? Why? What parameters/aspects did you consider to make the choice? [1 cubic foot ≈ 28.3 litres]
- The following figures show different types of cylinders. What is different in each of these cylinders?
- What is the area of the curved surface of a cylinder?
Let the radius of the circular base of the cylinder be r; let its height be h. Suppose the cylinder is cut as shown in the figure below. We get a rectangle. - Savitri had to make a model of a cylindrical kaleidoscope for her science project. She wanted to use chart paper to make the curved surface of the kaleidoscope (see the figure). What would be the area of chart paper required by her, if she wanted to make a kaleidoscope of length 25 cm with a 3.5 cm radius? You may take π = 227.