Question

A cube is made by arranging 64 cubes having side of 1 cm, find the total surface area of the cube so formed.

Answer 1

Hint:In this question we are going to solve this problem by equating the volumes of given 64 cubes and the volume of new formed cube by arranging the 64 cubes in such a way to make a larger cube. Further we use the formula for total surface area of the cube and the Volume of the cube. And the formula is listed below.

Formula used:Volume of the cube =\[{a^3}\]
Total surface area of the cube = \[6{a^2}\]
 where ‘a’ is the side of the cube.

Complete step-by-step answer:
 We are going to find out the volume of 64 cubes having side of 1cm,
Before that, we are going to calculate the volume of cube of side 1cm,
By using the formula of volume of cube stated above, we can find the volume of cube of side is 1cm
Volume of the cube =\[{a^3}\]
We are going to substitute a=1, then we get,
Volume of one cube of side 1cm = \[{\left( 1 \right)^3} = 1\]\[c{m^3}\]
Now, we are going to multiply the above volume by 64 to get the volume of 64 cubes having side of 1cm
Then, we get
Volume of 64 cubes having side of 1cm =\[64 \times 1c{m^3} = 64c{m^3}\]----------------- [1]
Now, we are going to calculate the volume of the newly formed cube.
Let us assume that \[a\] be the side of a new cube.
Volume of cube formed = \[{a^3}\]------------------ [2]
Then using the formula of Volume of cube, we are going to calculate the volume of the newly formed cube by arranging 64 cubes.
Now we are going to use the concept that 64 cubes are used to make a one single cube so the volume of 64 cubes of side 1cm will be equal to the one single cube.
Then we have,
Volume of 64 cubes of side 1 cm = Volume cube formed.
Now, we are going to put the values in equation [1] and [2] in above expression. Then we get\[{a^3} = 64c{m^3}\]
Now, we are going to calculate the side of cube formed by taking cube root on both sides,
\[ \Rightarrow a = 4cm\]------------- [3]
Hence we found the side of the newly formed cube by arranging 64 cubes having side of 1 cm.
Now we are going to find the total surface area.
Total surface area of new formed cube is = \[6{a^2}\]
Now we are going to substitute the value of a, Then we get,
Total surface area = \[6{\left( 4 \right)^2}\]=\[6 \times 16\]
Therefore, Total surface area = 96

Hence, the total surface area of cube so formed is \[96c{m^2}\]

Note:We can easily solve these types of problems if we know the formula for given shape. Also one must not confuse volume and total surface area. The region or space of the plane figure or object is called area. The quantity of space contained by an object is called volume. Plane figures have area while solid shapes have volume. Area describes the amount of space enclosed, whereas volume determines the capacity of solids. The measurement of area is done in square units, which can be centimeters, yards and so on. On the contrary, the volume is measured in cubic units. Shapes having two dimensions, i.e. length and width have area. As against this, shapes with three dimensions, i.e. length, width and height, have volume.