Solid

one solid. three shadows. find the angle.

turn one solid until its three shadows are a circle, a square and a triangle

A shadow is not a copy of an object. It is one question asked of it -- what do you look like from exactly here -- and the answer throws away everything along that one direction. Ask three such questions along three perpendicular axes and a single solid can answer circle, square and triangle at the same time: a cylinder with a ridged roof, cut by two slanted planes, does precisely that, and all three answers inscribe the same square. Turning the solid changes every shadow at once, because each shadow is a silhouette along an axis fixed to the room rather than to the object. There are exactly two orientations, mirror images of each other, where all three answers come out clean.

Concepts: geometry, perspective.

An interactive you play with one thumb, on Wundr.