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
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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?
This problem could work well as a 'poster' - a visual challenge placed where students will see it. Or presented at the end of a lesson as something to try to solve.
This problem can worked well as something short and closed, but there is also an opportunity to invite questions which open up beyond the initial challenge. Finding the area of a square in a quadrant or squares fitted in the space between either of the two squares in the main problem and the circle.
The approach suggested above indicates one route to extension within this context, or for another challenge fitting squares into shapes try Squirty. The problem Semi-square offers another opportunity to work out areas of squares inside circles.
Tilted Squares could be an excellent and accessible challenge for slightly less experienced students.