The problem is posted here: http://nrich.maths.org/5673 I tried subbing in values to get the biggest area: perimeter ratio possible. I used an arbitrary value, 12, for perimeter. For the triangle, I took out the hypotenuse and so A = 1/2(x)(y) where x+ y = 12. Its area is biggest when x = y = 6 so it has an area of 36. For the rectangle I took out the longer side of the rectangle and so 2x + y = 12. A = x * y. I differentiated and found that its area is biggest when x = 3 and y = 6, giving a area of 18. Now for the circle I was lost on how to approach it since the arc length and area vary in a different manner. I just used a semicircle and guessed that its area would be the largest so I used pi * r = 12. and A = (pi * r^2)/2. r = 12/pi and I subbed that in for area and came up with 22.9183 for area. I have no idea if my logic behind this is correct so I would appreciate any feedback/solutions to this problem. Thanks!!