### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

# Three Cubes

##### Age 14 to 16 Challenge Level:

1. The distance between centres of two adjacent faces of a cube is $8$cm.

What is the side length of the cube?

2. I can make an equilateral triangle by cutting off the corner of a cube.

If the area of the largest equilateral triangle I can create in this way is $140$cm$^2$, what is the side length of this cube?

3. A third cube has an edge length of $12$cm.

At each vertex, a tetrahedron with three mutually perpendicular edges of length $4$cm is cut away.

Can you find the volume and the surface area of the remaining solid?
•
•