### 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?

### Reach for Polydron

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

### 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.

# Tetra Inequalities

##### Stage: 5 Challenge Level:

Try a proof by contradiction and use the Triangle Inequality which says that a triangle can be constructed with three given segments for sides if and only if the sum of the lengths of any two exceeds the length of the third. (For example the lengths $2$, $3$ and $7$ cannot make the sides of a triangle because $2+3 < 7$.)

One more hint, one of the edges of the tetrahedron must be the longest and, without loss of generality, you can label this edge $AB$. Now, if you are using a proof by contradiction, what can you say about the 3 edges meeting at $A$ and similarly about the three edges meeting at $B$?