Dihedral angle calculation in pyramid with square base ABCD. Let OABCD be a pyramid with square base ABCD such that the angle between lateral edge OA and OB,the angle between lateral edge OB and OC,the angle between lateral edge OC and OD,the angle between lateral edge OD and OA is $\frac{\pi}{4}$.Then what is the angle between the planes OAB and OBC,what is the angle between the planes OAB and ABC,and what is the volume of the pyramid. EDIT 1:The sides OA=OB=OC=OD=1 unit I dont have working knowledge of pyramids.Can someone please guide me how can i find the angle between lateral faces and angle between a lateral face and the base if the angle between the lateral edges is given.

The angle between two planes $\alpha$ and $\beta$, having a line of intersection $r$, is defined as the angle formed by two lines $a\subset \alpha$ and $b\subset\beta$, both perpendicular to $r$ at the same point $M$.

In your case, to find the angle between $OAB$ and $OBC$ you can for instance draw the altitudes $AM$ and $CM$ of those two faces: then the angle is $\angle AMC$. To compute it you can find the lengths of $AM$, $CM$ and $AC$ (by some standard trigonometry) and then use the cosine law to get the angle.

The angle between $OAB$ and $ABCD$ is even easier: if $OK$ is another altitude of face $OAB$, then the angle is just $\angle OKH$, where $H$ is the center of square base.