Given a cube with edge 10, find the angle between the diagonal and the plane of the base.

Let’s construct the diagonal AC1 of the cube. The projection of the AC1 diagonal onto the plane of the base of the cube is the diagonal AC of the base of AВСD. Then the linear angle CAC1 is the angle between the diagonal AC1 and the plane of the base of the cube.

The cube has all the faces of squares, then in a right-angled triangle ABC, according to the Pythagorean theorem, AC ^ 2 = AB ^ 2 + BC ^ 2 = 100 + 100 = 200.

AC = 10 * √2 cm.

In a right-angled triangle ACC1, tgCAC1 = CC1 / AC = 10 / (10 * √2) = √2 / 2.

Angle CAC1 = arctan (√2 / 2).

Answer: The angle between the diagonal and the base plane is arctan (√2 / 2).