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?

24
12
16
8

Answer 1

Answer:

♠ The correct option is D.

Explanation:

♦ A side of big cube of 12 cm is cut into smaller cube of side 3 cm. So, one side of big cube equals 4 sides of smaller cube.

♣The big cube is cut into 4 x 4 x 4 = 64 small cubes.

♣ There are 8 original corners - 3 painted sides  

♣ There are 12 edges on the original cube, each will give two cubes with two painted sides - 24 in all

♣ There were 6 faces, each with four cubes with one painted side - 24 in all

So, the number with no painted sides is 64 - 8 - 24 - 24 = 8

Answer 2

The total number of smaller cubes having none of their faces painted is 8 Cubes.

Given:

A cube of side 12cm is painted red on its face.

To Find:

The number of smaller cubes having none of their faces painted.

Solution:

Let n denotes the number of smaller cubes.

The volume of a large cube = n * Volume of smaller cubes.

12³ = n×3³

⇒ n = (3³×4³)/3³

⇒ n = 4³ = 64

On one edge $\frac{12}{3} = 4$ parts will be present.

∴ On one faces , 4 * 4 = 16 cubes will be present

On 8 edges, the smaller cubes are painted in three faces = 8

Smaller Cubes with two faces painted = (4*2) *3 = 24

Smaller Cubes painted on one faces = 4 *6 =24

Smaller cubes with painted faces= 24+24+8 = 56

Total no of smaller cubes  having none of their faces painted

= Total no of smaller cubes- smaller cubes with their faces painted

= 64-56

= 8

Therefore, the total number of smaller cubes having none of their faces painted is 8 Cubes.

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