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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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Pericut

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).

Bound to Be

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

Bounded area $ABCD$ is a square of side 1 unit. Arc of circles with centres at $A, B, C, D$ are drawn in.

Prove that the area of the central region bounded by the four arcs is:

$(1 + \pi/3 + \sqrt{3})$ square units.

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