Regular polygons and circles

Two polygons fit together so that the exterior angle at each end of their shared side is 81 degrees. If both shapes now have to be regular could the angle still be 81 degrees?

Weekly Problem 23 - 2008
A triangle has been drawn inside this circle. Can you find the length of the chord it forms?

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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

Can you find a relationship between the area of the crescents and the area of the triangle?

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Weekly Problem 5 - 2006
How many times does the inside disc have to roll around the inside of the ring to return to its initial position?

The diagram shows two semicircular arcs... What is the diameter of the shaded region?