P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
Find the perimeter and area of a holly leaf that will not lie flat
(it has negative curvature with 'circles' having circumference
greater than 2πr).
The ten arcs forming the edges of this "holly leaf" are all arcs
of circles of radius 1 cm. At all the spiky points the circles
touch each other tangentially. All the straight lines join the
centres of the circles they pass through and the four triangles at
the corners have angles of 45, 45 and 90 degrees.
Find the length of the perimeter of the holly leaf and the area
of its surface.
This is a 10-spike holly leaf. What is the perimeter of a
16-spike holly leaf of the same sort?
Examine some real holly leaves and you will find that they don't
lie flat. This shape is different from a real holly leaf in so far
as it does lie flat on a flat surface (or plane). See '
Giant Holly Leaf' for an extension of this problem to a more
realistic holly leaf which has negative curvature.