Rotating Triangle

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

Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

An Unusual Shape

Stage: 3 Challenge Level:

Stuart tethered his goat to one end of a $20$-foot length of rope and attached the other end of the rope to a hook fixed to the side of his shed.

When he came back the goat had eaten all the grass within its reach.
Here is a picture showing the region eaten by the goat:

Where did Stuart fix the rope?

What was the area grazed by the goat?

Could the goat have eaten more grass if the rope had been attached to a different point on the shed?

Experiment with different lengths of rope. Where should the hook be positioned each time to allow the goat to eat the maximum amount of grass?