# Perfect Shapes

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Introduction

Perfect Shapes A perfect shape is said to be one that has the same area and perimeter. (1) Find the area and perimeter of the rectangle below. (2) Is the rectangle perfect? (3) Investigate perfect rectangles (4) Extend your investigation into other shapes. Perfect Shapes A perfect shape is defined to be one in which its perimeter is equal to its area. Rectangles I will begin by assuming that two of the sides of the rectangle are of length 5: A 5 by 1 rectangle is therefore not perfect and we need to investigate others: A 5 by 2 rectangle is not perfect either. I will tabulate further results in my quest to find a perfect 5 by something rectangle: Rectangle Length Width Area Perimeter Perimeter - Area 5 1 5 12 7 5 2 10 14 4 5 3 15 16 1 5 4 20 18 -2 5 5 25 20 -5 5 6 30 22 -8 5 7 35 24 -11 5 8 40 26 -14 5 9 45 28 -17 5 10 50 30 -20 Table 1 Looking at Table 1 it can be seen that no perfect shape has been found. ...read more.

Middle

7 28 22 -6 4 8 32 24 -8 4 9 36 26 -10 4 10 40 28 -12 This is interesting in that it shows that a 4 by something perfect rectangle is a square. Let us consider a general square to see if any other perfect square exists: A graph to illustrate this relationship might be useful: Graph 1 We can see from Graph 1 that in two places: and . The zero solution makes no sense but this confirms that if a square is 4 by 4 then it is perfect. As there is only one non-zero crossing point it also illustrates that there is only one perfect square. Let us return to the algebra briefly: Which confirms the result from the graph and proves that a 4 by 4 square is the only one that exits. It might be interesting to see if this pattern is true for other regular polygons, is there only one perfect equilateral triangle? Is there only one perfect equilateral pentagon? These questions I will attempt to address later. So far in this investigation the length of a rectangle has been fixed, it will be useful to relax this condition and consider the more general case of a perfect rectangle: Suppose that the length and ...read more.

Conclusion

#NUM! 25 #NUM! B C D E F G 2 y x h A P 3 Length of the equal sides Length of the base Perpendicular height Area of Triangle Perimeter of triangle Perimeter -Area 4 7 1 6.98 3.49 15 11.51 5 7 2 6.93 6.93 16 9.07 6 7 3 6.84 10.26 17 6.74 7 7 4 6.71 13.42 18 4.58 8 7 5 6.54 16.35 19 2.65 9 7 6 6.32 18.97 20 1.03 10 7 7 6.06 21.22 21 -0.22 11 7 8 5.74 22.98 22 -0.98 12 7 9 5.36 24.13 23 -1.13 13 7 10 4.90 24.49 24 -0.49 14 7 11 4.33 23.82 25 1.18 15 7 12 3.61 21.63 26 4.37 16 7 13 2.60 16.89 27 -10.11 Note the lack of sign change in the Area - Perimeter column suggesting that perfect 4, 4, x and 6, 6, x isosceles triangles do not exist. There are two sign changes in the 7, 7, x table and so it seems that two perfect isosceles triangles exist with these dimensions. Let me investigate this further: To be completed... let me know if you would like a copy of the completed version.......... Page 1 of 1 ...read more.

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