All its 12 edges are equal in length. So, a cube is a cuboid in which length = breadth = height

The volume of a cube will be measured by the cube of the length of its side. Denoting the volume of a cube by V and its length L.

So,  V = Lᶟ,  L = 3√ V

The surface area of a cube is the sum of the areas of its six square faces or 6 X (length of an edge)². Denoting the surface area by S,

S = 6L² and  L = √(S/6)

The area of the four walls (lateral surface area) = 4 X (length of an edge)² = 4L² 

Since a cube is a special cuboid in which l = b = h = L

the length of a diagonal of a cube = √ l² + b² + h²

                                      =  √ L² + L² + L² 

                                      =  √3L²  =  √3 L

so, the length of a diagonal = √3 L     

Example.) The length of a diagonal of a cube is 15√3 cm. find the (a) length of an edge of the cube, (b) the volume of a cube, and (c) the surface area of the cube.

Ans.)  let the length of each edge of a cube is = a cm

As per the rules length of the diagonal = √3a cm

So, as per the given condition-  √3 a  =  15√3,       

                                       a = 15 cm 

so, the length of the edge of the cube is 15 cm…………………..(a)

the desired volume of the cube is = (length of a side)ᶟ

                                     = (15 cm)ᶟ =  3375 cmᶟ ………………..(b)

the surface area of the cube = 6 X (length of a side)²

                                 =  6 X (15)² = 1350 cm² ……………..(c)