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?
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.
- 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
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