Areas of Geometric shapes with the same perimeter (challenge question)

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      Areas of Geometric shapes with the same perimeter 


      The problem linked above is part of a very rich tradition of problems looking to maximize the area enclosed by a shape with fixed perimeter. Only three shapes are considered here because the problem is

      difficult for more irregular shapes. For example, of all triangles, the one with fixed perimeter

      and largest area is the equilateral triangle whose side lengths are all P/3 but this is difficult to

      show because it is not easy to find the area of triangle in terms of the three side lengths (though

      Heron’s formula accomplishes this). Nor is it simple to compare the area of two triangles with

      equal perimeter without knowing their individual areas. For quadrilaterals, a similar problem

      arises: showing that of all rectangles with perimeter the one with the largest area is the

      square whose side lengths are P/4 is a good problem which students should think about. But

      comparing a square to an irregularly shaped quadrilateral of equal perimeter will be difficult.


      For this problem, very explicit shapes have been chosen aiming at providing an opportunity for

      students to practice using their knowledge of different geometric formulas for area and

      perimeter. The teacher, in order to reinforce part (d), may wish to draw examples of a hexagon

      and octagon (both sharing the same perimeter as the other figures). The central idea is that as

      the number of sides of the polygon increase, the polygon looks more and more like the circle

      and in particular its area is getting closer to the area of the circle.


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