Regular polygons and circles

The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.

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.

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?

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?

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.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.