The Fencing Problem

Regular polygons have the largest areas

This shows that the closer the values of the height and width the larger the area. I made many more of these calculations and created a graph.

This graph reinforces that 250 x 250 is the rectangle with the largest area which shows that the shapes with the largest area are those with equal sides.

Number of Sides

Area

3

48111.11

4

62500

5

68800

6

72168.72

7

73890.13

8

75444.17

9

76318.81

0

76942.09

Conclusion

Regular polygons have larger areas than irregular polygons, so a rectangle could have the same perimeter as a square but it wouldn't have as much area. Regular polygons with lots of sides have more area than regular polygons with less sides, so a square could have the same perimeter as a hexagon but the hexagon would have more area. Because regular polygons with more sides have a larger area and a circle has an infinite number of sides a circle is the shape with the largest area. But because the relationship between number of sides and area is a curve rather than a straight line a circle has almost the same area as a 360 sided regular polygon or any other shape with a high number of sides.