Polyhedra

problem
Sliced

Age

14 to 16

Challenge level

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?
problem
Shadow play

Age

5 to 7

Challenge level

Here are shadows of some 3D shapes. What shapes could have made them?
problem
Tetrahedra tester

Age

14 to 16

Challenge level

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?
problem
A chain of eight polyhedra

Age

7 to 11

Challenge level

Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?
problem
Cut nets

Age

7 to 11

Challenge level

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?
problem
Tetra perp

Age

16 to 18

Challenge level

Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular if and only if $AB^2 +CD^2 = AC^2+BD^2$. This problem uses the scalar product of two vectors.
problem
Pythagoras for a tetrahedron

Age

16 to 18

Challenge level

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.
problem
The dodecahedron

Age

16 to 18

Challenge level

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?
problem
Dodecawhat

Age

14 to 16

Challenge level

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.
problem
Classifying solids using angle deficiency

Age

11 to 16

Challenge level

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry