## Cube

Cube is a solid three-dimensional figure with 6 square faces, eight vertices, and 12 edges, in geometry. It is also said to be a regular hexahedron. In simple words, it is a solid box-shaped object with six identical square faces. In the image below, L stands for length, B stands for width, and H stands for height. There are many applications of cubes in day-to-day life as well as in mathematical problems, thereby making it one of the most important geometric shapes.

Here are a few examples of cubes in daily life:

Dice

Rubik’s Cube

Ice and sugar cube

Building blocks

Boxes

Note: Since the cube is a solid figure, it has three dimensions, while the plane figures have only two dimensions.

## Properties of Cube:

It's got all its faces in a square shape.

All faces or sides are of equal dimensions.

The plane angle of the cube is the right angle.

Each face meets the other four faces.

Each of the vertices has three faces and three edges.

The edges are parallel which are opposite each other.

### Face Diagonal of a Cube Formula:

A cube has six faces. So, it’ll have twelve face diagonals (two diagonals on each face). The diagonals form the hypotenuse of a right-angled triangle in the Cube, making each leg be formed by one edge. Hence, the length will be (a√2).

## Diagonal of a Cube Formula:

Now, let's discuss the length of the diagonal of the cube formula. The main diagonal of the Cube is the one that cuts through the center of the Cube; the diagonal of the Cube's face is not the main diagonal. The main diagonal of every Cube can be determined by multiplying one side's length with the square root of 3 (this is also known as the body diagonal of a cube formula).

As discussed earlier, in a cube, all sides have equal length.

So, let's say that each side has a length x.

Then, formula to find diagonal of cube = √3x.

## Diagonal of a Cube Formula Derivation:

Let’s consider cube with face diagonal ‘a’ & body diagonal ‘d’.

In the above diagram, we have a right triangle with sides a, c and d.

By Pythagoras theorem, we have

a^{2} + c^{2} = d^{2} - (Equation 1)

We already discussed the formula of face diagonal i.e., c = a√2

By substituting the value of c in Equation 1, we get,

⇒ a^{2} + (a√2)^{2} = d^{2}

⇒ a^{2 }+ (2a^{2}) = d^{2}

⇒ 3a^{2} = d^{2}

⇒ d = √(3a^{2})

⇒ d = a√3

Therefore, Diagonal of cube = a√3.

## Examples based on Diagonal of a Cube Formula:

Example 1: Find the diagonal of the cube with the given side 10 cm?

Ans: Side is given as, x = 10 cm

We know that,

The diagonal of the cube is √3x

So, Diagonal of cube = √3 (10)

We know that the value of √3 is 1.734

Therefore, the diagonal of a cube = 1.734 x (10)

So, Diagonal of cube = 17.34 cm

Example 2: If the edge of a cube measures 5 cm, find the length of a diagonal.

Ans: Side is given as, x = 5 cm

We know that,

Diagonal of cube = √3x

Diagonal of cube = √3 (5)

We know that the value of √3 is 1.734

Therefore, the diagonal of a cube = 1.734 x (5)

So, Diagonal of cube = 8.67 cm.

### Other Formulas of Cube are as Follows:

The other formulas mentioned for cube here are defined based on their Surface areas (TSA), (LSA) Lateral surface areas, and Volume(V):

Total Surface Area = 6 x Side^{2}

Lateral Surface Area = 4 x Side^{2}

The volume of cube = Side^{3}

Perimeter of cube = 12 x Side

### The Difference Between Cube and Cuboid:

A cube is a three-dimensional shape recognized as an XYZ plane. There are six faces, eight vertices, and twelve edges in it. All the cube faces have equal dimensions and are square in shape.

A polyhedron with six faces, eight vertices, and twelve edges is a cuboid. The cuboid faces are parallel to each other. But not all the cuboid faces are dimensionally equivalent.

Major differences between both are as follows:

The cube sides are the same, but the cuboids are different.

In shape, the cube's sides are square, but the cuboid is rectangular.

All the cube diagonals are the same, but the cuboid has the same diagonals for parallel sides only.

These are the key points that are the reasons why there is a difference between the formula of diagonal of cube and cuboid.

### Why is a Cube a Special Kind of Cuboid?

A cube can be considered as a special kind of cuboid where all the cuboid faces are of equal length. There are six faces in the cuboid, which are rectangles. If the rectangles have the same sides, they become squares, and eventually, the cuboid becomes a cube.

### Conclusion

These notes are prepared in such a way that it's easy to understand the concepts of cubes and cuboids. We observed that a 3D solid cube has six sides. There are 8 vertices and 12 edges so that at one vertex, three edges meet. Here, we also derived the formula for the diagonal of a cube and compared the features of cubes and cuboids. So, you must go through the solved examples given here to understand and learn the concepts and their applications well.

1. How Will You Find the Diagonal of a Cube?

Ans: The main diagonal of the cube is the one that cuts through the center of the cube; the diagonal of the cube's face is not the main diagonal. The main diagonal of every cube can be determined by multiplying the length of one side with the square root of 3.

2. What is the Diagonal of a Cube?

Ans: The diagonal of the cube is a segment that joins two points that are not the endpoints of the edge. Whereas the two diagonals of the square are of the same length, the cube has diagonals of two different lengths, the shorter ones on the square and the longer ones passing through the center.

3. What is the Face Diagonal of a Cube?

Ans: A face diagonal of a polyhedron is a diagonal on one of the faces, in contrast to space diagonal passing through the interior of the polyhedron.