# Diagonal of a Cube Formula – Definition, Derivation, Examples

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## What Is the Diagonal of a Cube Formula?

The formula to calculate the length of diagonal of a cube (body diagonal) is 3 times the length of one of its sides. In mathematical terms, if “s” represents the length of a side, then the diagonal “d” can be expressed as $d = s\sqrt{3}$.

A cube is a three-dimensional solid figure with 6 congruent square faces, 8 vertices, and 12 equal edges. All its edges are equal in length. This indicates that a cube has equal length, breadth, and height and that every one of its sides is a square.

The diagonal of a cube is the line segment that connects any pair of non-adjacent sides of the cube. The main diagonal of a cube is the one that goes through the center of the cube; the face diagonal of a cube is not its primary diagonal.

There are two kinds of diagonals of a cube.

• Face Diagonal of a Cube

There are six square-shaped faces on a cube. The non-adjacent vertices of each face are connected by two diagonals. Therefore, there are actually 12 face diagonals.

• Body Diagonal of a Cube

The body diagonals connect the two opposite vertices of the cube by passing through the internal structure of the cube. As a result, there are 4 diagonals crossing across the body. It is also known as “space diagonal,” or “internal diagonal,” or “main diagonal.”

## Diagonal of a Cube Formula

Let’s see the formulas to find the diagonal of a cube.

Example 1: Calculate the length of the face diagonal of a cube whose each side is 8 inches long.

Length of the side of the cube = a = 8 inches

Length of face diagonal of the cube $= \sqrt{2a}$

$\Rightarrow d = \sqrt{2} \times 8$

$\Rightarrow d = 1.41 \times 8$

$\Rightarrow d = 11.28$ inches

Example 2: Calculate the length of the body diagonal of a cube with side-length of 9 inches.

Length of the side of the cube = a = 9 inches

Length of body diagonal of the cube $= \sqrt{3a}$

$d = \sqrt{3} \times 9$

$d = 1.73 \times 9$

Length of body diagonal of the cube $= 15.57$ inches

## Number of Diagonals in a Cube

We know that a cube has 6 faces. Each face has 2 diagonals connecting their non-adjacent sides.

Number of face diagonals $= (2 \times 6) = 12$

Additionally, there are 4 diagonals running across the body of the cube.

Number of body diagonals $= 4$

Therefore, the total no. of diagonals in a cube can be calculated as

$12 + 4 = 16$

## Derivation of Diagonal of a Cube Formula

The formula for the diagonal of a cube is calculated using the Pythagoras theorem. The cube diagonal formula is usually referred to as the body diagonal formula.

Consider a cube whose each side length is “a” units. If we draw any face diagonal, we can form a right-angled triangle with legs of “a” units and face diagonal as the hypotenuse.

Face diagonal $= \sqrt{a^{2} + a^{2}} = \sqrt{2a^{2}} = \sqrt{2a}$ units

Similarly, we will get a right-angled triangle with the body diagonal as the hypotenuse, side of the cube as one leg, and the face diagonal as the other leg.

Side length = a

face diagonal $= \sqrt{2a}$

By Pythagoras theorem, we get

$a^{2} + (\sqrt{2a})^{2} = d^{2}$

$\Rightarrow a^{2} + 2a^{2} = d^{2}$

$\Rightarrow 3a^{2} = d^{2}$

$\Rightarrow d = \sqrt{3a^{2}}$

$\Rightarrow d = \sqrt{3a}$

Length of the Body diagonal $= d = \sqrt{3a}$

## Uses of Diagonal of a Cube Formula

1. Determining Cube Size: The formula helps figure out how long the diagonal is within a cube, which is like measuring the longest diagonal in a rectangular box.
2. Visualizing 3D Shapes: Understanding cube diagonals gives you a way to imagine how lines can go from one corner of the cube to the farthest corner, helping you see how 3D shapes fit together.
3. Understanding Big Structures: Architects and builders use these diagonals to make sure big structures, like bridges or buildings, are strong and stable when they have cube-like parts.
4. Creating Cube Designs: If you want to build something using cubes, like a robot or a piece of furniture, knowing how diagonals work will help you make sure everything fits together just right.

## Facts about the Diagonal of a Cube Formula

• The diagonal is the longest line that can be drawn within a cube, connecting two opposite corners (vertices).
• If you need to travel from one corner of a cube to its opposite corner, the body diagonal (space diagonal)  is the shortest path.

## Conclusion

In this article, we explored the formulas for calculating the face and body diagonals of a cube, crucial for determining distances between vertices. These concepts are pivotal in geometry and spatial reasoning. For enhanced understanding, let’s work through examples and engage in MCQs.

## Solved Examples on Diagonal of a Cube Formula

Example 1: Find the face diagonal of a cube with sides of 7 units.

Solution:

Length of the side of the cube = 7 units

Length of face diagonal of the cube $= \sqrt{2a}$

Length of face diagonal of the cube $= \sqrt{2} \times 7$

Length of face diagonal of the cube $= 1.41 \times 7$

Length of face diagonal of the cube $= 9.87$ units

Example 2: What will be the body diagonal of a cube with sides of 11 inches?

Solution:

Length of the side of the cube = 11 inches

Length of body diagonal of the cube $= \sqrt{3a}$

Length of body diagonal of the cube $= \sqrt{3} \times 11$

Length of body diagonal of the cube $= 1.73 \times 11$

Length of body diagonal of the cube =19.03 inches

Example 3: The body diagonal of a cube is 4√3 units. What will be the length side of the cube and the length of the face diagonal of the cube?

Solution:

Length of body diagonal of the cube $= \sqrt{3a}$

$\sqrt{3a} = 4\sqrt{3}$

Length of side of the cube =4 units

Length of the side of the cube =4 units

Length of face diagonal of the cube $= \sqrt{2a}$

Length of face diagonal of the cube $= \sqrt{2} \times 4$

Length of face diagonal of the cube $= 1.41 \times 4$

Length of face diagonal of the cube $= 5.64$ units

## Practice Problems on Diagonal of a Cube Formula

1

### What will be the length of the face diagonal of a cube with a side of 12 inches?

15 inches
16.50 inches
16.92 inches
20.76 inches
CorrectIncorrect
Length of face diagonal of the cube $= \sqrt{2a} = \sqrt{2} \times 12 = 1.41 \times 12 = 16.92$ inches
2

### What will be the length of the body diagonal of a cube with a side of 6.1 inches?

10 inches
10.55 inches
10.38 inches
8.60 inches
CorrectIncorrect
Length of body diagonal of the cube $= \sqrt{3a} = \sqrt{3} \times 6.1 = 1.73 \times 6.1 = 10.55$ inches
3

### Find the length of the side of a cube if the face diagonal is given as $9\sqrt{2}$ inches.

9 inches
12.69 inches
18 inches
15.57 inches
CorrectIncorrect
Length of face diagonal of the cube $= \sqrt{2a} = 9\sqrt{2}$
Thus, $a = 9$ inches

If a face diagonal lies on a face of a cuboid with dimensions of the face as a × b, then its length is given by $\sqrt{a^{2} + b^{2}}$.

The body diagonal of a cuboid $= \sqrt{l^{2} + b^{2} + h^{2}}$

The diagonal of a cube is a line segment connecting two opposite corners of the cube, passing through the center. It helps visualize the longest distance within the cube.

The face diagonal of a cube is found using the formula: face diagonal = side length $\times \sqrt{2}$.

This helps calculate the diagonal within a face of the cube.

The body diagonal of a cube is given by the formula: body diagonal = side length $\times \sqrt{3}$.

This formula calculates the longest diagonal that passes through the cube’s body.

There are 12 face diagonals and 4 body diagonals.