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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?

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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.

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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

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why use this problem?

The problem introduces an attractive generalisation of Pythagoras' theorem to 3D and gives good practice in the use of the area formula for a triangle and the cosine rule. It involves the use of Pythagoras' theorem in 2D and some algebraic manipulation.

An entirely different approach to solving the problem involves vectors.

Possible approach

Pre-planning of how to tackle the problem involves asking:

Key questions

What information is given?
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?

Possible support

Try the problem Rectangular Pyramids.

Possible extension

Prove the result using vectors.
Generalise the result to a cosine rule for a tetrahedron.