Polyhedra

problem
Favourite

Tetra Perp

Age

16 to 18

Challenge level

2 out of 3

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
Favourite

Pythagoras for a Tetrahedron

Age

16 to 18

Challenge level

3 out of 3

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

Reach for Polydron

Age

16 to 18

Challenge level

1 out of 3

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.
problem

Magnetic Personality

Age

7 to 16

Challenge level

1 out of 3

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?
problem

Lighting Up Time

Age

7 to 14

Challenge level

1 out of 3

A very mathematical light - what can you see?
problem

Tetra Square

Age

14 to 18

Challenge level

2 out of 3

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.
problem

Rhombicubocts

Age

11 to 14

Challenge level

2 out of 3

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices does each solid have?
problem

Sliced

Age

14 to 16

Challenge level

2 out of 3

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

A Mean Tetrahedron

Age

11 to 14

Challenge level

3 out of 3

Can you number the vertices, edges and faces of a tetrahedron so that the number on each edge is the mean of the numbers on the adjacent vertices and the mean of the numbers on the adjacent faces?
problem

Face Painting

Age

7 to 11

Challenge level

2 out of 3

You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.