Regular polygons and circles

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?

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

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

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?

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?

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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

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.

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