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# Pythagoras for a Tetrahedron

**Why use this problem?**

**Key questions**

**Possible support**

**Possible extension**

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The problem introduces an attractive generalisation of Pythagoras' theorem to 3D.

There is the opportunity to explore different solution techniques. Some methods include:

- Using the area of a triangle formulae and Pythagoras's theorem.
- Using Heron's formula.
- Using vectors.

What information is given?

In what different ways could we find the area of a triangle?

What can be deduced from the geometrical properties?

What are the unknowns and what notation shall we use?

What equations can we write down for the areas of the four triangular faces of the tetrahedron?

What equations can we write down involving the lengths of the edges of the tetrahedron?

How can we put all this together to get the required result?

Try the problem Rectangular Pyramids.

Try using different methods to solve the problem.

Generalise the result to a cosine rule for a tetrahedron.

**STEP Support Programme Foundation Assignment 5** asks a different question about the same tetrahedron.

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.

Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.