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Volume of a Cube

Volume of a Cube: 
Since all of the edges of a cube are of equal length, there is no need to differentiate between the length, width and height.  The volume of a cube is determined by multiplying the length of three edges of the cube.  Let "a" define the length of an edge of the cube.  

Then the volume formula is defined as:

V = a × a × a
This formula is also written as V = a³.

If a = 5 feet, then the volume is found like this:

V = (5)³ = 5 × 5 × 5 = 125:  Since each value is a measurement of feet, the volume is therefore 125 cubic feet. 

Example 1: Find the volume of a solid cube shape box whose side is 13m.
Solution:
Given that: side a = 3m
Volume of the cube is therefore:

V = a³
V = 13³
V = 2197 m3

Example 2: Find the measure of volume of a cube for the given side length is 20 inches?
Solution:
Given that: Side a = 20 inches
Volume of the cube is therefore:

V = a³
V = 20³
V = 8000 in3
Therefore, volume of a cube is 8000 in3.


Example 3: Find the volume of a cube for the given side length 12.5 meter?
Solution:
Given that: Side a = 12.5 m
V = a³
V = 12.5³
V = 1953.125 m3
Therefore, the volume of the cube is 1953.125 cubic meter.


Example 4: 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 the sum of the surface areas of the smaller cubes?
Solution:
The volume of the larger cube = 53 = 125 cm3.
The volume of each of the smaller cubes = 1
3 = 1 cm3. Therefore, one would get 125 smaller cubes out of the lager cube.
The surface area of the larger cube = 6a
2 = 6(52) = 6x25 = 150 cm2.
The surface area of each of the smaller cubes is:  6(1
2) = 6 cm2.
The total surface area of all the smaller cubes is: 125
× 6 = 750 cm2.
Therefore, the ratio is: 150: 750 = 1: 5


Example 5: A large cube is formed from the material obtained by melting three smaller cubes of 3, 4 and 5 cm side. What is the ratio of the total surface areas of the smaller cubes and the large cubes?
Solution:
Volume of the large cube = (33 + 43 + 53) = 216 cm3.
Let the edge of the large cube be
a.

a3 = 216
a = 6 cm

Required ratio = total surface area of the smaller cubes / the surface area of the larger cube
Required ratio = (6 × (32+42+52)) : (6 × 62) = 25:18

Example 6:  A cube has a surface area of 54 square centimeters. What is the volume of the cube?
Solution:
The surface area of the cube is:
SA = 6a2
Thus: a2 = SA/6 = 54/6 = 9 and a = 3 cm

The volume of the cube is:

V = a3
V = 33
V = 27 cm3


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