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The shape with the maximum area for a given perimeter is a circle.

A circle with perimeter (circumference) 30m has a radius = 30/(2*pi) = 15/pi m.

Then, its area is pi*(15/pi)2 = 225/pi = 71.62 sq metres.

If the question requires the shape to be polygonal, then the greater the number of sides (and so the closer the shape is to a circle) the greater the area enclosed.

Finally, if you are limited to squares and rectangles, the maximum area is for a square.

A square with perimeter 30 m has sides of length 30/4 = 7.5 metres. A square with sides of 7.5 m has an area of 56.25 sq metres (considerably smaller than the circle).

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Q: If you have 30 meters of fencing how can you arrange it so you have as much land as possible?
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