Diagonal of a Cube Formula: Definition, Properties and Sample Questions

The diagonal of a cube formula is \(\begin{array}{l}\sqrt{3}x\end{array}\). The diagonal of a cube helps calculate the lengths of the diagonals (which are the face diagonals and the body diagonals). The diagonal of a cube is the line segment which connects two non-adjacent cube vertices. It can be derived by using the Pythagoras Theorem. Cubes are solid three-dimensional figures with 6 square faces, 12 edges, and eight vertices.

A cube is said to be a regular hexahedron. In geometry, the length, breadth and height of a cube are all equal, which makes the figure symmetrical in shape. It is usually represented in an XYZ plane. The faces of a cube are in a squared shape with equal dimensions. The angles of a cube are found to be at a right angle. And, the opposite edges of a cube are parallel to one another.

Key Terms: Cube, Symmetry, Square, Diagonal, Right-angle, Vertex, Sides, Height, Three-Dimensional Geometry

What is a Cube?

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A cube is a regular hexahedron which has 6 square faces, 12 edges and eight vertices. Cubes are three-dimensional solid shapes which have equal length, width, and height with each face as a square.

Diagonal of a Cube

The diagonal of a cube, as can be seen from the above image as well, cuts via the centre of the cube. However, the diagonal of a face of a cube is not considered the main diagonal. Typically, the main diagonal of a cube can be determined by the help of multiplying the length of one side with the square root of 3.

What is Diagonal of a Cube?

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The diagonal of a cube can be defined as a line segment, in the cube, which connects any two non-adjacent vertices. There are mainly two types of diagonals in a cube because it is in three-dimension:

Face Diagonal of a Cube

In Face-Diagonal, a cube has 6 square-shaped faces with 2 diagonals on either face that joins the non-adjacent vertices. Thus, a cube has 12 face diagonals.

Body Diagonal of a Cube

Body diagonals typically connect the opposite vertices of a cube which pass via the body of the cube. Thus, a cube here has 4 body diagonals which pass via the cube connecting the opposite corners.

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Diagonal of a Cube Formula

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The Diagonal of a Cube Formula can be denoted by:

The primary diagonal of a Cube cuts through the centre of the Cube; the diagonal of the Cube's face is not the main diagonal. The main diagonal of a cube can be determined by the help of multiplying the length of one side with the square root of 3 (it is also called the body diagonal of a cube).

Diagonal of a Cube

Diagonal of a Cube Derivation

With the help of Pythagoras theorem, we get a² + r² = d² (… equation 1)

Now, we are aware that the face diagonal is √2a so, 'r' can be substituted by √2a, meaning 'r' = √2a

By replacing the value of r in equation 1, 

⇒ a² + (√2a)² = d²

⇒ a² + 2a² = d²

⇒ 3a² = d²

⇒ d = √(3a²)

⇒ d = √3a

Hence, the diagonal of a cube formula is = √3a

Properties of a Cube

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There are several properties of a cube. Some of them are:

• A Cube has 6 faces in square shape.
• The dimensions of the faces and sides of a cube are equal.
• All the angles in a Cube are at right angle.
• Any face of a cube meets the other four at each of the edges.
• The vertex of a Cube has three edges and three faces.
• Opposite edges of a Cube are parallel to one another.

Things to Remember

• The diagonal of a cube formula is \(\begin{array}{l}\sqrt{3}x\end{array}\).
• The diagonal of a cube calculates the lengths of the diagonals (Simply, face diagonals and the body diagonals).
• Cubes are known as solid three-dimensional figures that has 6 square faces, 12 edges, and eight vertices.
• There are two diagonals of a cube, Face Diagonal of a Cube and Body Diagonal of a Cube.