# Diagonal of a Cube Formula

The diagonal of a cube formula helps in calculating the lengths of different diagonals of a cube, namely, the face diagonals and the body diagonal. A cube is a three-dimensional solid figure, also known as the square solid that has edges of all the same measure, meaning that it has the length, width, and height equal, and each of its faces is a square. Let us understand the diagonal of a cube formula for measuring the lengths of both face and body diagonals using examples.

## What is Diagonal of a Cube Formula?

Cube is one of the important geometric shapes as this 3D shape has all 12 equal edges and we use cube-shaped shapes in real life also. Some of the real-life examples of a cube are ice cubes, sugar cubes, Rubik's cubes, playing dice, etc. The diagonal of a cube formula helps in finding the measures of the face diagonals and the main diagonals. Each face diagonal forms the hypotenuse of the right-angled triangle formed.

- Face Diagonals of a cube: A cube has 6 square-shaped faces. There are 2 diagonals on each face joining the non-adjacent vertices. Thus we have 12 face diagonals.
- Body Diagonals of a cube: The main diagonals connect the opposite vertices of the cube passing through the body of the cube. Thus we have 4 body diagonals.

The formula to calculate the length of a face diagonal of a cube is given as,

**Length of each face diagonal of cube = √2 x units**, where x = Length of each side of a cube

The formula to calculate the length of a space or body diagonal of a cube is given as,

**Length of each body diagonal of a cube = √3 x units**, where x = Length of each side of a cube

## Derivation of Diagonal of a Cube Formula

The diagonal of a cube formula can be derived by considering the side of the cube as x, a face diagonal as r, and body diagonal as d. Once we join the diagonals, right triangles are formed with sides x, r, and d.

By using the Pythagoras theorem, we have x^{2} + r^{2} = d^{2} (let's keep this as equation 1)

We already know that the face diagonal is √2 x so, r = √2 x

By substituting the value of r in equation 1, we get

x^{2} + (√2 x )^{2} = d^{2}

x^{2} + 2x^{2} = d^{2}

3x^{2} = d^{2}

d = √3x^{2}

d = √3 x

Therefore, the diagonal of a cube formula = √3 x units

**Break down tough concepts through simple visuals.**

## Examples Using Diagonal of a Cube Formula

**Example 1: Calculate the length of the diagonal of a cube whose each side measures 5 units.**

**Solution:**

Using the diagonal of a cube formula,

Length of body diagonal of a cube = √3 x units.

= √3 × 5 units = 8.66 units

Therefore, the length of the diagonal of a given cube = 8.66 units.

**Example 2: The length of the diagonal of a cube is 8√3 units. Find the length of each side.**

**Solution:**

Using the diagonal of a cube formula,

Length of diagonal of a cube = √3 x units = 8√3 units

x = 8√3/√3 = 8 units

Therefore, the length of each side of the given cube = 8 units.

**Example 3: Find the length of the face diagonal of a cube when the side measures 6 units. Use the face diagonal formula of a cube. **

**Solution: **

Using the face diagonal of a cube formula,

Length of each face diagonal of cube = √2 x units

= √2 × 6 units = 8.49 units.

Therefore, the length of the face diagonal of a cube is 8.49 units.

## FAQs on Diagonal of a Cube Formula

### What is the Diagonal of a Cube Formula?

The diagonal of a cube formula helps is measuring the diagonal of the square-shaped faces of the cube. The formula of the diagonal of a cube is used in finding - the face diagonal and the main diagonal of the cube. The face diagonal is the diagonal of the side of the cube i.e. 6 faces = 12 diagonals. While the main diagonal crosses the center of the cube.

### What is the Formula to Calculate the Face Diagonal of a Cube?

The formula to calculate the length of face diagonal of a cube is given as,

Length of each face diagonal of cube = √2 x units, where x = Length of each side of a cube

### What is the Formula to Calculate the Diagonal of a Cube?

The formula to calculate the length of body or main diagonal of a cube is given as,

Length of body diagonal of a cube = √3 x units, where x = Length of each side of a cube

### Using the Diagonal of a Cube Formula, Find the Length of the Diagonal with the Side Measuring 8units.

Using the diagonal of a cube formula,

Length of body diagonal of a cube = √3 x units.

= √3 × 8 units = 13.86 units

Therefore, the length of the diagonal of a given cube = 13.86 units.