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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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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?

Weekly Problem 4 - 2006

Stage: 4 Challenge Level: Challenge Level:1
A roll of adhesive tape is wound round a central cylindrical core of radius $3 \; \text{cm}$. The outer radius of a roll containing $20 \; \text{m}$ of tape is $4 \; \text{cm}$. What is the outer radius of a roll containing $80 \; \text{m}$ of tape?

This problem is taken from the UKMT Mathematical Challenges.

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