P is the midpoint of an edge of a cube and Q divides another edge
in the ratio 1 to 4. Find the ratio of the volumes of the two
pieces of the cube cut by a plane through PQ and a vertex.

What is the surface area of the tetrahedron with one vertex at O
the vertex of a unit cube and the other vertices at the centres of
the faces of the cube not containing O?

Cubestick

Age 16 to 18 Challenge Level:

Pythagoras Theorem will give you all the important lengths here.

If you know the lengths of 3 sides of a triangle you can find the angles!