Pythagoras' theorem

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

A rectangular piece of paper is folded. Can you work out one of the lengths in the diagram?

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance up the wall which the ladder can reach?

A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket ?.

Weekly Problem 2 - 2009
The 16 by 9 rectangle is cut as shown. Rearrange the pieces to form a square. What is the perimeter of the square?

How does the perimeter change when we fold this isosceles triangle in half?

What is the perimeter of this unusually shaped polygon...

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?