Cube 

A cube is one of the types of 3-dimensional figures, i.e., consisting of height, breadth, and length; it has six equal square faces, where all angles are right angles and equal side lengths. 

1.0Basics of Cubes

As mentioned earlier, a cube shape is a three-dimensional geometric figure that has 6 square faces of equal lengths, 12 edges, and 8 vertices. It comes under one of the most symmetrical three-dimensional shapes, with all angles at right angles and all faces of equal size, that is, length and breadth. 

It is to be noted that a cube is a special case of a rectangular prism (Cuboid) where all three dimensions (length, width, and height) are the same.

2.0Cube Properties

• Equal Dimensions: A cube has equal length, width, and height. This is the defining property that distinguishes a cube from a general rectangular prism.
• Face-Centred Symmetry: Every face of a cube has a centre point, and the cube exhibits face-centred symmetry, meaning that rotating a face around its centre will produce an identical result.
• Geometric Properties: The centre of a cube is equidistant from all its faces. The diagonals of any face of the cube divide the square face into two equal right triangles.
• Symmetry: A cube has high symmetry. It is symmetric along several axes. It has 24 rotational symmetries, meaning it can be rotated in multiple ways and still look the same.
• Diagonals: The diagonal of a cube is defined as the line that joins two opposite vertices. A cube has two kinds of diagonals: space diagonal and face diagonal.

3.0Formulae Related to Cube

1. Cube Area: the area or the surface area of the cube refers to the total or the curved area of all six or four (in the case of hollow square)square faces, respectively. The formula for both areas of the cube with side let ‘a’ is:

Total Surface Area of a cube $=6a^2$

Curved Surface area of a cube $=4a^2$

2. Cube Volume: The volume measures the three-dimensional space present within a geometric figure. The volume of a cube is the amount of space it occupies. It is given by the formula:  

Volume of cube $(V)=a^3$

3. Diagonal of a Cube: The diagonal of a cube refers to the line segment joining two opposite vertices (non-adjacent vertices). The length of the diagonal d of a cube is expressed as: 

Length of diagonal $(d)=a\sqrt{3}$

4.0Problems on Cube

Problem 1: The edge of a cube is 8 cm. Find the volume and surface area of the cube.

Solution: given that the edge of a cube ‘a’ = 8cm 

Volume of cube $(V)=a^3$

$=8^3=512c{m}^3$

Surface area of a cube $=6a^2$

$=6\times 8^2=384c{m}^2$

Problem 2: Three small cubes, each with an edge length of 4 cm, are melted and combined to form one large cube. What is the edge length of the large cube?

Solution: let the edge of small cubes be a = 4 

Let the edge of the big cube be A =?

The volume of three cubes = volume of the resultant cube 

$3a^3=A^3$

$A^3=3\times 4^3$

$A=\sqrt{3\times 4^3}=4\sqrt{3}cm$

Problem 3: A hollow cube with only a bottom has an outer edge length of 10 cm and an inner edge length of 6 cm. The cube is made of a thin material, and both the outer and inner surfaces need to be painted. Find the total surface area that needs to be painted (both inside and outside).

Solution: let the inner edge be ‘a’ and the outer edge be ‘A’ 

Outer surface to be painted = outer curved surface area + bottom 

$=4A^2+A^2=5A^2$

$=5\times 10^2=500c{m}^2$

Inner surface area to be painted = Inner curved surface area + bottom 

$=4a^2+a^2=5a^2$

$=5\times 6^2=180c{m}^2$

Total surface area that needs to be pained = Outer surface to be painted + Inner surface area to be painted $=500+180=680c{m}^2$