A solid cube of side 2a has surface area 24a^2. What is the total surface area of all the pieces after n symmetrical cuts?
A cube of side 2a has surface area 24a^2. When it is cut parallel to a surface, through the centre, two cuboids of sides 2a, 2a and a are formed. The total surface area of one of these cuboids is 4(2a*a) + 2(2a*2a) = 16a^2. Therefore the total of both pieces is 32a^2. When these cuboids are cut again, symmetrically, the area increases to 40a^2.
No. of cuts Area/a^2
0 24
1 32
2 40
.... .....
n T(n)
Is this an A.P. with common difference 8?
Can you prove it?