Regular polygons and circles

Weekly Problem 50 - 2012
The diagram shows a regular dodecagon. What is the size of the marked angle?

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

Weekly Problem 52 - 2012
An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?

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

Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.

Thinking of circles as polygons with an infinite number of sides - but how does this help us with our understanding of the circumference of circle as pi x d? This challenge investigates this relationship.

Can you reproduce the design comprising a series of concentric circles? Test your understanding of the realtionship betwwn the circumference and diameter of a circle.

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

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?