Problem:
Pyramid OABCD has square base ABCD, congruent edges OA,OB,OC, and OD, and ∠AOB=45∘. Let θ be the measure of the dihedral angle formed by faces OAB and OBC. Given that cosθ=m+n, where m and n are integers, find m+n.
Solution:
Let P be the foot of the perpendicular from A to OB. The pyramid's symmetry implies that P is also the foot of the perpendicular from C to OB. Without loss of generality, we may assume OP=1, from which AP=PC=1,OB=OA=2, and BP=2−1 follow. Two applications of the Pythagorean Theorem now give
AB2=AP2+BP2=4−22
and
AC2=2⋅AB2=8−42
The measure of the dihedral angle determined by faces OAB and OBC is the same as that of ∠APC. Use the Law of Cosines to obtain
