area and perimeter
If you think about it long enough, the shape with the largest area for a given perimeter is the... what? Find the area of different shapes for perimeter, say, 12cm. Note: You might need to use a bit of Pythagoras and trigonometry of you're looking at polygons with lots of sides.
What about if I have two shapes of the same size and put them together, so they can share part of their perimeter? Are your results still the same?
What happens if I put lots of shapes together? Which shape is the best then?
What about the gaps in between the shapes? Do they matter?
How can you judge the shape with the maximum area?
You could use a ratio, such as A:P, or A/P. Then the shape with the highest ratio A/P is the winner.
They they use this idea in medical biology to determine whether cells could be cancerous. The word cancer derives from the greek word for crab, because cancerous cells are crab-like, or star-like, in shape (see image on the right).
How about investigating the ratio A/P for star-like shapes, or irregular shapes? What do you notice?
How about investigating these ideas in 3-D?