# Find the angle between the diagonal of a cube and the diagonal of one of its faces.

**Solution:**

Let us consider a cube with side length = a.

Length of diagonal of cube on one of its faces = a√2

Length of diagonal of cube = a√3

To find the angle between the, use law of cosines

The law of cosines is given by

⇒ c^{2} = a^{2 }+ b^{2 }- 2ab cos C

Where c = length of side c, b = length of side b, a = length of side a

C = angle opposite to c

⇒ c^{2} = (a√2)^{2} + (a√3)^{2} - 2 (a√2)(a√3) cos C

On simplification,

⇒ cosC = 2a^{2}/(a√2 × a√3)

= 2/(√2 × √3)

= √2/√3

C = cos^{-1}(√2/√3)

Therefore, the angle between the diagonals is cos^{-1}(√2/√3).

## Find the angle between the diagonal of a cube and the diagonal of one of its faces.

**Summary:**

The angle between the diagonal of a cube and the diagonal of one of its faces is cos^{-1}(√2/√3).

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