Show that the edges AD and BC of a tetrahedron ABCD are mutually
perpendicular when: AB²+CD² = AC²+BD².
In this article, we look at solids constructed using symmetries of
This problem provides training in visualisation and representation
of 3D shapes. You will need to imagine rotating cubes, squashing
cubes and even superimposing cubes!
Think about the bond angles occurring in a simple tetrahedral
molecule and ammonia.
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?
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
A description of how to make the five Platonic solids out of paper.
Can you work out the dimensions of the three cubes?
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.
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. . . .
An irregular tetrahedron has two opposite sides the same length a
and the line joining their midpoints is perpendicular to these two
edges and is of length b. What is the volume of the tetrahedron?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Prove that in every tetrahedron there is a vertex such that the
three edges meeting there have lengths which could be the sides of