# Cube Shape

A cube is a regular convex polyhedron.

It is a special cuboid.

You will be introduced to a set of formulas for surface area and volume of a cube.

You will have learned how to solve problems on cube with examples.

It is easy to understand.

## Cube

A cube is a polyhedron bounded by six congruent square faces that are placed at right angles to each other. Three edges of faces meet at a point is known as vertex of the cube.

A cube is a special type of cuboid.

A cube showing its faces and edges.

Each squared-face is called the surface of the cube. These surfaces are congruent square. Therefore, there are six congruent surfaces of a cube. Each adjacent surface is perpenducular to each other. By meeting the surfaces to each other, it produces several edges and vertices. The point at which three edges meet is called vertex of the cube.

A cube must have

- 6 congruent faces
- 12 edges of equal length
- 8 vertices.

A cube is a special solid object.

Thus a cube is also known as

- a regular polyhedron
- an equilateral cuboid
- a regular hexahedron
- a right rhombohedron
- a uniform polyhedron
- a square parallelopiped
- a trigonal trapezohedron
- a regular square prism
- one of the platonic solids.

### Cube Example

## Area of Cube

A cuboid is formed by squares that are conguent to each other. Thus the length, width and height of a cube are of equal length.

A cube showing its faces and edges

### Method one

Let a be the length of the dimension of a cube.

It is known to us, the surface area of cuboid = 2(ab + bc + ca) square unit

Since a cube is also a cuboid, its length = width = height.

So, set a = b = c.

∴ if the area of cube is A,

A = 2(ab + bc + ca) square unit

or, A = 2(a.a + a.a + a.a) square unit

or, A = 2(a^{2}+a^{2}+a^{2}) square unit

or, A = 2.3a^{2} square unit

∴ A = 6a^{2} square unit

### Method two

A cube consists of 6 surfaces and each surface is a square and congruent to each other.

Let the side of a square be of length a.

∴ the area of 1 surface = a^{2} square unit

As a cube is formed with 6 squared congruent surfaces, the area area of the cube is the sum of six surfaces.

∴ if the area of cube A, the area formula for the cube is

A = (a^{2} + a^{2} + a^{2} + a^{2} + a^{2} + a^{2}) square units

∴ A = 6a^{2} square units

If the length of the each edge of a cube is a and the area of total surfaces is A,

A = 6a^{2} square units

## Volume of Cube

The volume of a cube is the base area times the height, that is

### Method one

Let the length of the edge be a.

∴ Area of base a.a = a^{2} square unit

∴ Volume of cube = (Area of base × height) cubic unit

or, Volume of cube = a^{2}.a cubic unit

∴ Volume of cube = a^{3} cubic unit

### Method two

Let the length, width and height are the a, b and c respectively.

∴ Volume of cuboid = abc cubic unit

Sine a cube is a special cuboid, its length = width = height ie, a = b = c

setting, a = b = c

Volume of cube =a.a.a cubic unit

∴ volume of cube = a^{3} cubic unit

volume of cube = (length × length × length) cubic unit

if the length of each edge is a and volume is V,

V = a^{3} cubic unit

## Diagonal of Cube

### Method one

Firstly, we can determine the diagonal of base.

The each side of the base is a. The triangle is formed by the two sides and diagonal is a right triangle.

Let the length of each side be a and diagonal of the base be k. So, Pythagorean theorem implies

k^{2} = a^{2} + a^{2}

or, k^{2} = 2a^{2}

or, k = √2a^{2}

∴ k = a√2

In figure, d is the diagonal.

Again, a triangle is formed by the three sides a, d and a√2 that is a right triangle whose hypotenuse is d.

A cube is showing its diagonal.

By the Pythagorean theorem,

d^{2} = a^{2} + (a√2)^{2}

or, d^{2} = a^{2} + 2a^{2}

or, d^{2} = 3a^{2}

or, d = √3a^{2}

∴ d = a√3

### Method two

Let the dimension of a cuboid be a, b, c and a diogonal d. Then

d = √a^{2} + b^{2} + c^{2}

As a cube is also a cuboid, it implies length = width = height.

that is, for a cube a = b = c.

∴ if the diagonal of the cube is d,

d = √a^{2} + a^{2} + a^{2}

or, d = √3a^{2}

∴ d = a√3

If the length of the each edge of a cube is a and a diagonal is d,

d = a√3 unit