GATE CSE 2016 Set 1 | Question: GA05

Question

A cube is built using $64$ cubic blocks of side one unit. After it is built, one cubic block is removed from every corner of the cube. The resulting surface area of the body (in square units) after the removal is ________.

1. $56$
2. $64$
3. $72$
4. $96$

Comments

If in place of surface area, if the question was to find the new volume, what would have the answer been?

96-8=88?

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@s_tripathi NO. The volume will be 64 - 8 = 56

Answer 1

Cube is built using $64$ cubic blocks of side one unit. It means it is a $4\times 4\times 4$ Cube.

Surface area of each face $ = 4\times 4$ sq. units

And number of faces in a cube $= 6$, Thus total surface area $ = 6\times 4\times 4 = 96$ sq. units.

A cube contains $8$ corner pieces (containing $3$ visible sides), and if these are removed then inner ones are visible, So, When a corner piece is removed $3$ new faces of $1$ sq. unit are visible and thus removal doesn't change surface area.

Hence, Option $(D)$ is correct.

Comments

Since when we cut the subcube out of main cube , three surfaces get removed but three surface also gets added..Hence no effect on surface area..

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yes, that I understood, but my doubt is in the first part, i.e. how to calculate the total surface area in the first place.

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Assume unit cube as a point.

Volume of a cube = $(side) ^3$

Here, volume = 64 $point^3$ , so side = 4 points.

Answer 2

A cube is built with 64 blocks of 1 unit side. Hence each side will have surface area $4\times4= 16$ square units.

if we remove one block from each corner, the resulting surface area will reduce by 4 square units. Hence surface area of each side is 12 square units.

Total surface area of 6 sides is $12\times6= 72$. But removal of a block from each corner creates an surface area of 3 square units.

Since there are 8 corners, the total surface area becomes $72 +3\times8 =  96$.

Hence answer is D.

Comments

this is a very good explanation..!!!

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72+3*8=96

Answer 3

the surface area of the body will remain unchanged as when a cube is removed, it exposes three faces, which makes the number of exposed faces same as before removal.
 If 64 cubic blocks are used to make the cube, it must be 4 x 4 x 4 blocks
So, surface area of the body before removal = surface area of the body after removal = 6 * side * side = 6 * 4 * 4 = 96.
 
Thus, D is the correct choice.