Regular polygons and circles

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

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

Can you find the link between these beautiful circle patterns and Farey Sequences?

Investigate constructible images which contain rational areas.