Dihedral angle in the cube

Let us see the normal section at work in a concrete case.

Example — Dihedral angle in the cube

In a cube $ABCDEFGH$ with side $\ell=2$ consider the plane $NMBD$, where $N$ and $M$ are the midpoints of two upper edges. The dihedral angle between the plane $NMBD$ and the base $ABCD$ is measured by drawing a plane $\perp$ to the line of intersection $BD$ and calculating the angle between the traces.

By Pythagoras’ theorem: $BD = 2\sqrt{2}, \qquad NM = \sqrt{5}, \qquad DN = \sqrt{5}.$ The dihedral angle turns out to be $\alpha = \arctan(\sqrt{2}) \approx 70{,}5°.$

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Topics: Synthetic geometry in space
Concepts: Dihedral angle · Normal section
Skills: Calculating · Synthetic geometry