I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius $4$ cm and touches one side of the box, the middle sized one has radius $9$ cm and touches two sides of the box and the largest one touches three sides of the box. What is the radius of the largest one?

This problem by R B J T Allenby appeared in the Sunday Times on 12.11.00.

Sine, cosine, tangent. Mathematical reasoning & proof. Circle properties and circle theorems. Cartesian equations of circles. Regular polygons and circles. Similarity and congruence. Sine rule & cosine rule. 2D shapes and their properties. Creating and manipulating expressions and formulae. Pythagoras' theorem. Polyhedra.