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

Rotating Triangle

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

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:

$\sqrt{37} \; \text{cm}$. The cross section of the $20 \; \text{m}$ tape has area $\pi(4^2 - 3^2) \text{cm}^2 = 7\pi \text{cm}^2$ Therefore, the $80 \; \text{m}$ tape should have a cross-section area $28\pi \; \text{cm}^2$. Hence, the outer radius of the $80 \;\text{m}$ roll will be approximately $\sqrt{37} \;\text{cm}$

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.

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