Polyhedra

problem
Favourite

Octa-Flower

Age

16 to 18

Challenge level

1 out of 3

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?
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?