The angle between two face diagonals of a cube

Question

Take a cube. On one face draw a diagonal, and on an adjacent face draw a diagonal, so that the two diagonals meet at a shared corner of the cube.

What is the angle between the two diagonals at the corner where they meet? The tempting answer is 90 degrees, since the faces are at right angles. Is that right?

Accepted Answer

60 degrees. The triangle. Call the shared corner A. The first diagonal runs from A to the opposite corner B of one face; the second runs from A to the opposite corner C of the adjacent face. Now look at the segment BC. It is also a face diagonal: B and C are opposite corners of a third face of the cube, the one that touches both of the first two faces but does not contain A. So AB, AC and BC are all face diagonals of the same cube and all have the same length. Coordinates. Put A at (0, 0, 0) with the cube of side 1. Take B = (1, 1, 0) (diagonal of the bottom face) and C = (1, 0, 1) (diagonal …

The angle between two face diagonals of a cube

Your answer

Discussion

Learn the concepts

Related questions