Question

A cube of side 12 cm is painted red on all the faces and then cut into smaller cubes, each of side 3 cm. What is the total number of smaller cubes having none of their faces painted?

• A. 8
• B. 27
• C. 16
• D. 64

Hint:

When dealing with cubes cut into smaller cubes, focus on the interior cubes for those with no painted faces.

Answer 1

The Correct Option is A

The large cube has a side length of 12 cm. The total number of smaller cubes is found by dividing the volume of the large cube by the volume of each smaller cube. The volume of the large cube is: \[ \text{Volume of large cube} = 12^3 = 1728 \, \text{cm}^3 \] Each smaller cube has a side length of 3 cm, so the volume of each smaller cube is: \[ \text{Volume of smaller cube} = 3^3 = 27 \, \text{cm}^3 \] Thus, the number of smaller cubes is: \[ \text{Number of smaller cubes} = \frac{1728}{27} = 64 \] Now, to find the number of smaller cubes that have no faces painted, we observe that the cubes in the interior of the large cube will not have any faces painted. The interior cubes form a smaller cube with side length: \[ \text{Side length of interior cube} = 12 - 2 \times 3 = 6 \, \text{cm} \] The number of smaller cubes in the interior of the large cube is: \[ \left( \frac{6}{3} \right)^3 = 2^3 = 8 \] Thus, the number of smaller cubes that have no faces painted is a. 8.