What is Cube?
A cuboid is a cube if its length, breadth and height are equal.
In a cube, all the faces are squares which are equal in area and all the edges are equal. A dice is an example of a cube.
Volume of a Cube (V) = (edge)^{3} = a^{3}
Total surface Area of a Cube (S) = 6(edge)^{2} = 6a^{2}
Diagonal a Cube (d) = √3(edge) = √3a
Where a = edge
Problems on Volume and Surface Area of Cube:
1. If the edge of a cube measures 5 cm, find (i) it volume, (ii) its surface area, and (iii) the length of a diagonal.
Solution:
(i) volume = (edge)^{3}
= 5^{3} cm^{3}
= 125 cm^{3}
(ii) Surface area = 6(edge)^{2}
= 6 × 5^{2} cm^{2}
= 6 × 25 cm^{2}
= 150 cm^{2}
(iii) The length of a diagonal = √3(edge)
= √3 × 5 cm.
= 5√3 cm.
2. If the surface area of a cube is 96 cm^{2}, find its volume.
Solution:
Let the edge of the cube be x.
Then, its surface area = 6x^{2}
Therefore, 96 cm^{2} = 6x^{2}
⟹ x^{2} = \(\frac{96 cm^{2}}{6}\)
⟹ x^{2} = 16 cm^{2}
⟹ x = 4 cm.
Therefore, edge = 4 cm.
Therefore, the volume = (edge)^{3}
= 4^{3} cm^{3}
= 64 cm^{3}.
3. A cube of edge 2 cm is divided into cubes of edge 1 cm. How many cubes will be made? Find the total surface area of the smaller cubes.
Solution:
Volume of the bigger cube = (edge)^{3}
= 2^{3} cm^{3}
= 8 cm^{3}.
Volume of each of the smaller cubes = (edge)^{3}
= 1^{3} cm^{3}
= 1 cm^{3}
Therefore, the number of smaller cubes = \(\frac{8 cm^{3}}{1 cm^{3}}\)
= 8
The total surface area of a smaller cube = 6(edge)^{2}
= 6 × 1 cm^{2}
= 6 cm^{2}
Therefore, the total surface area of the eight smaller cubes = 8 × 6 cm^{2 }= 48 cm^{2}.
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