Regular polygons and circles

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

In LOGO circles can be described in terms of polygons with an infinite (in this case large number) of sides - investigate this definition further.

Make an estimate of how many light fittings you can see. Was your estimate a good one? How can you decide?

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

These four touching circles have another circle hiding amongst them...

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

See if you can anticipate successive 'generations' of the two animals shown here.

Weekly Problem 53 - 2012
ABCDEFGHI is a regular nine-sided polygon (called a 'nonagon' or 'enneagon'). What is the size of the angle FAE ?

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?