Square Pegs

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

The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

Isosceles

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

A Chordingly

Age 11 to 14Challenge Level
Prove that the area of the annulus is ${1\over 4}\pi AB^2$ where $AB$ is a tangent to the inner circle.