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

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Areas of Geometric shapes with the same perimeterÂ

The problem linked above is part of a very rich tradition of problems looking to maximize the area enclosedÂ 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.