Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?

If you move the tiles around, can you make squares with different coloured edges?

Make some loops out of regular hexagons. What rules can you discover?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

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

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Find the perimeter and area of a holly leaf that will not lie flat (it has negative curvature with 'circles' having circumference greater than 2πr).