Pythagoras' theorem

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

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.