Regular polygons and circles

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

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?

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

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?

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

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.

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.

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.

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.