### 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:

This solution came from Ruth from Manchester High School for Girls. Well done Ruth!

Let the tetrahedron's vertices be $A$, $B$, $C$ and $D$ and the longest side be $AB$. If you assume that there is not a vertex where the three sides meeting at it could be the sides of a triangle, we must have $AC + AD < AB$ and $BC + BD < AB$ (otherwise the sides meeting at $A$ or $B$ could be the sides of a triangle). Therefore

$$AC + AD + BC + BD< 2 AB$$.Now since $ABC$ and $ABD$ are both triangles, we must have $AC + BC > AB$ and $AD + BD > AB$. Therefore

$$AC+ AD + BC + BD > 2 AB$$. This contradicts (1), so the initial assumption must be wrong. There is at least one vertex (one of $A$ or $B$) where the three sides meeting at it could be the sides of a triangle.