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Just Rolling Round

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 of P?

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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?

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Giant Holly Leaf

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).


Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.