The original puzzleTwo diagonals on the sides of a cube meet in a corner. What is the size of the angle between them ? The SolutionThis is a real beauty !     The answer to the original question is not too difficult to find,          but it leads to all sorts of new questions concerning volume. The AnalysisEach of the 2 given diagonals, on adjacent faces of a cube, has 2 ends -     one end at which they meet, and another open, "loose" end.If a line is drawn between these 2 "loose" ends,     it will be another (third) diagonal on a face adjacent to the other 2 faces.These 3 diagonals form an equilateral triangle,     giving us the answer to our puzzle. MathematicsThis equilateral triangle of diagonals, acts as the base of a tetrahedron,     whose sides are 3 right-angle isosceles triangles,          each occupying half of a cube face, 1.5 cube faces in all.There are 4 such tetrahedrons occupying the total surface of the 6 faces of the cube.     There also seems to be a fifth tetrahedron, this one perfectly regular,          on the inside, enclosed in the 4 others.How is the volume of the cube distributed among these 5 tetrahedrons ? PedagogyIt might be fun for the children to cut up a soft-plastic (or styrofoam) cube,along the six diagonals of the cube (one on each face),to remove 4 pyramids from the corners(all built on an equilateral base with 3 isosceles right-angle sides),and discover another pyramid "hidden" inside(perfectly equilateral, base and sides),larger than the other 4 exterior pyramids.