Question

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

Short answer

The angle between the diagonal of a cube and the diagonal of one of its sides is approximately 35.26 degrees.

Step-by-step solution

Define the parameters and draw the cube

First, let's denote the edge of the cube as 'a'. Now, visualize a cube and draw it with its edges and diagonals.

Find the Face Diagonal

Use the Pythagorean theorem across one face of the cube to calculate the length of the face diagonal. In a right triangle formed by one edge of the cube, the face diagonal, and another edge, the hypotenuse would be the face diagonal. By Pythagorean theorem, the face diagonal \(d_f\) is given by \(\sqrt{a^2 + a^2} = a\sqrt{2}\)

Find the Cube Diagonal

Now, to find the diagonal of the cube, we need to use the Pythagorean theorem again. This time, in a right triangle formed by an edge, the face diagonal (already calculated in step 2), and the hypotenuse would be the cube diagonal. Hence, cube diagonal \(d_c\) is given by \(\sqrt{a^2 + (a\sqrt{2})^2} = a\sqrt{3}\)

Find the Required Angle

In a right triangle formed by a side of the cube, the diagonal of the cube \(d_c\) and the diagonal of the face \(d_f\), the required angle is the angle between \(d_c\) and \(d_f\). This angle \(ø\) can be found using the cosine formula, which in our case would be: \(Cos ø = \frac{d_f}{d_c} = \frac{a\sqrt{2}}{a\sqrt{3}} = \sqrt{\frac{2}{3}}\). The angle \(ø\) is therefore given by the arccos of this value, which is approximately 35.26 degrees.