# Cube

Introduction

What Is A Cube?

Cube is a 3-dimensional hexahedron, made up of 6 squares of same sides.  Each cube has six square faces, eight vertices and twelve edges. In 3 dimension, some faces share a common boundary called edge. It is the bounding line of the faces. Vertex is a corner where two or more faces and edges segments meet. A face is a plain flat surface. The points given below define the structure of a cube:

1. Each face is connected to four vertices and four edges

2. Each vertex is connected to three edges and three faces

3. Each edge is in touch two faces and two vertex

Also, the length, breadth and height of a cube are the same because it is the three-dimensional figure of a square that has all the sides of the same length.

A cube can be called as a regular hexahedron, a square parallelepiped, an equilateral cuboid, a right rhombohedron, a regular square prism, a trigonal trapezohedron, etc.

Surface Area and Volume of a Cube.

Surface area is the covering of a three-dimensional object. The total surface area is the addition or sum of the area of all the surfaces of an object. For example, the amount of wrapping paper required to cover the surface of an object is the total surface area of an object.

Volume is the three-dimensional space occupied by any object. It is not just a surface but the whole space. For example, the amount of water a glass can take is called the volume of the glass.

The surface area of a Cube

Surface area is of one face of a cube is equal to the area of a square having sides the same as the cube. The total surface area of a cube is the flat region occupied by all the planes of the cube. A cube has six planes of squares. Now, we know that the area of a square is the square of its side. So for six surfaces, the area occupied by all the six planes is six times the area of the square, that is, 6 times a2

Therefore, the formula for the total surface area of a cube is:

The surface area of cube = 6a2.

The Volume of a Cube

The total internal space occupied by a cube is called the volume of a cube. The formula for the volume is the product of its length, breadth and height. Since, the length, breadth and height of a cube are equal to the side of the square that makes up the cube. Therefore. The volume of a cube is given by the given formula:

The volume of a cube = a x a x a.

= a3.

The volume of a cube is a cube of the length of its side.

# Length of diagonal of a cube

What is diagonal of a cube?

Diagonal of a cube are of types:

Space diagonal: The line connecting two vertices that are not in touch with the same faces of a cube is called diagonal. A cube can have 16 diagonal but all of the same length.

Face diagonal: It is the diagonal of the square face of a cube which is equal to the line connecting two opposite corner of a square. Face diagonal lie on the surface of the square having the same side.

### How to find the length of a diagonal?

The process of finding the length of the diagonal is more like Pythagoras theorem. Let the length be x, breadth be y and height be z.

Since, length (x), breadth (y) and height (z) are all same.

Let, x=y=z = a

(Face Diagonal)2 = (Length )2+ (Breadth)2

(Fc)2 = (x)2+ (y)2

(Fc)2 = (a)2+ (a)2

(Fc)2 = 2(a)2

$F_{c}$ = $\sqrt{2}$a

Therefore, the face diagonal is equal to $\sqrt{2}$a.

(Space Diagonal)2 = (Length )2+ (Breadth)2+ (Height )2

(Fs)2 = (x)2+ (y)2+ (z)2

(Fs)2 = (a)2+ (a)2+ (a)2

(Fs)2 = 3(a)2

$F_{s}$ = $\sqrt{3}$a

Therefore, the space diagonal is equal to $\sqrt{3}$a.

Properties of a cube.

1. It has six square faces

2. The side of all the square is the same.

3. The angle between any two surfaces or faces is always 90 degrees.

4. The edges of each square face are connected to for more square faces.

5. The opposite planes or faces are parallel to each other.

6. The opposite edges are also parallel to each other.