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Last modified on March 28th, 2023

The diagonal of a cube is the straight line connecting any of its two non-adjacent vertices. It is also called the body or space diagonal since it passes through the center of a cube. There are 4 body diagonals in a cube. Since the diagonal is a linear quantity, it is expressed in linear parameters such as mm, cm, m, in, or ft.

The formula is:

Let us have a cube with an edge of ‘a’ units.

∴ AB = BD = DC = a, as shown in the figure below.

The face diagonal is the hypotenuse of the right triangle △BDC.

∴ applying Pythagoras theorem, we get,

(BC)^{2} = (BD)^{2} + (DC)^{2}

(BC)^{2} = a^{2} + a^{2}, since BD = DC = a

(BC)^{2} = 2a^{2 } … (1)

∴ BC = √2a (here BC is the face diagonal)

So, the **face diagonal **of a cube **= √2a**

Also, in △ABC, the body diagonal AC is the hypotenuse.

∴ applying Pythagoras theorem, we get,

(AC)^{2} = (AB)^{2} + (BC)^{2}

Let AC = d

∴ d^{2} = (AB)^{2} + (BC)^{2}

d^{2} = a^{2} + 2a^{2 }, [replacing (BC)^{2} = 2a^{2 }from (1)]

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

d = √3a

Hence, we get the formula for the diagonal of a cube as **d = √3a**

We also get the formula:

**Face diagonal = √2a**, where a = edge

The **face diagonal** connects two non-adjacent vertices on a face. It is not the main diagonal of a cube. There are 12 face diagonals in a cube.

**Find the length of a diagonal of a cube whose edge is 6 cm.**

Solution:

As we know,

The diagonal of a cube means its body diagonal,

Body Diagonal (d) = ${\sqrt{3}a}$, here a = 6 cm

∴d = √3 × 6

= 10.39 cm

**Calculate the diagonals of a cube as given in the figure alongside.**

Solution:

As we know,**Body Diagonal (d) =** ${\sqrt{3}a}$, here a = 8 cm

∴ d = √3 × 8

= 13.86 cm**Face Diagonal =** ${\sqrt{2}a}$, here a = 8 cm

∴ d = √2 × 8

= 11.31 cm

**Calculate the diagonal of a 3×3 Rubik’s cube.**

Solution:

As we know,

Body Diagonal (d) = ${\sqrt{3}a}$, here a = 3 units

∴ d = √3 × 3

= 5.2 cm

Last modified on March 28th, 2023