### 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?

### Tetra Inequalities

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

### Pythagoras for a Tetrahedron

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.

# Reach for Polydron

##### Stage: 5 Challenge Level:

To discover the easy way to find this volume you can very quickly make for yourself a model of the tetrahedron. Take a square $ABCD$ of stiff paper and fold it along the diagonal $AC$ . Open it until the distance between $B$ and $D$ is equal to the length of the sides of the square and then the four vertices are the vertices of the tetrahedron with two equilateral faces and two isosceles faces. You can either stand this 'tetrahedron' on an equilateral face ( $ABD$ or $CBD$ ) as its base or on an isosceles face ( $ABC$ or $ADC$ ) as its base. One choice makes it easy to find the area of the base and the height and so to find the volume.