Can you say what angle is made by the two red lines drawn on two sides of the cube as shown in the illustration?

Construct a sphere around the vertex where the two lines meet. The spherical triangle cut off by the two marked planes and the plane of the triangle will have two sides of 45 degrees, with the angle between them of 90 degrees.  The law of cosines can be used to determine the side opposite the 90-degree angle, and that will be the angle sought (planar angles correspond to sides of the spherical triangle, and dihedral angles correspond to angles on the spherical triangle).

cos x = cos 45 * cos 45 + sin 45 * sin 45 * cos 90

cos x = (1/sqrt(2))^2 = 1/2

x = 60 degrees

Posted by Charlie on 2004-08-05 15:48:11