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.
Diagonal of a Cube
Formula
The formula is:
Diagonal of Cube Formula
Derivation
Let us have a cube with an edge of ‘a’ units.
∴ AB = BD = DC = a, as shown in the figure below.
Diagonal of a Cube Formula Derivation
The face diagonal is the hypotenuse of the right triangle △BDC.
∴ applying Pythagoras theorem, we get,
(BC)2 = (BD)2 + (DC)2
(BC)2 = a2 + a2, since BD = DC = a
(BC)2 = 2a2 … (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
∴ d2 = (AB)2 + (BC)2
d2 = a2 + 2a2 , [replacing (BC)2 = 2a2 from (1)]
d2 = 3a2
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.
Solved Examples
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
Calculate the diagonals of a cube as given in the figure alongside.