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Cut Cube

Find the shape and symmetries of the two pieces of this cut cube.

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Plane to See

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.

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Four Points on a Cube

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?


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!

Or you could use vectors and scalar products.