Regular polygons and circles

Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make an estimate.

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?

How efficiently can you pack together disks?

This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

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."