Student: Dinkal Airen

Q. How to solve volume and surface area questions. Can you explain the method. Here is the problem. A 5 cm cube is cut into as many 1 cm cubes as possible. What is the ratio of the surface area of the larger cube to that of the sum of the surface areas of the smaller cubes?


Answer :

Hi Dinkal, Thanks for your Question.

Summary : To solve problems related to volume and surface area, we need to visualize the shapes,  think through and solve as mentioned below.
Concepts:

  • Cube is a solid with six congruent square faces.<
  • A cube has six faces which are all squares, so each face has four equal sides and all four interior angles are right angles.
  • A cube has 12 edges. Because all faces are squares and congruent to each other, all 12 edges are the same length.

Method:  Say L is the length of any edge of a cube.
Then Volume of a Cube =  L * L * L   = L3

Surface Area of Cube : There are 6 faces of cube. Each face is square.
Surface area of one face = L * L = L2
Total Surface area of Cube =  6 * L * L = 6L2

Now let`s solve the problem you mentioned.
The volume of the larger cube = 5 * 5 * 5 = 125 cm3.  
The volume of each of the smaller cubes = 1 * 1 * 1 = 1 cm3 

As volume would remain equal as same large cube is cut into smaller cubes. 
Number of smaller cubes = 125/ 1 
So there would be 125 smaller cubes. 
The surface area of the larger cube = 6 * 5 * 5 = 150 cm2 
The surface area of each of the smaller cubes = 6 * 1 * 1 = 6 cm2
Therefore, surface area of all of the 125 smaller cubes = 125 * 6 = 750 cm2
Surface area of larger cube = 150 cm2
Surface area of all of the 125 smaller cubes = 750 cm2

Therefore, the required ratio of surface area of larger cube to all smaller cubes  = 150 : 750 = 1 : 5


Dinkal, Hope you got what you were looking for.  Keep browsing the blogs for more examples. Post the query on home page if you have further questions.
 

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