A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

Solution:

Side of a cube = 12cm (Given)

Find the volume of cube:

Volume of cube = (Side)3 = (12)3cm3= 1728cm3

Surface area of a cube with side 12 cm = 6a2 = 6(12) 2 cm2 …(1)

Cube is cut into eight small cubes of equal volume, say side of each cube is p.

Volume of a small cube = p3

Surface area = 6p2 …(2)

Volume of each small cube = (1728/8) cm3 = 216 cm3

Or (p)3 = 216 cm3

Or p = 6 cm

Now, Surface areas of the cubes ratios = (Surface area of bigger cube)/(Surface area of smaller cubes)

From equation (1) and (2), we get

Surface areas of the cubes ratios = (6a2)/(6p2) = a2/p2 = 122/62 = 4

Therefore, the required ratio is 4 : 1.