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  • Level: GCSE
  • Subject: Maths
  • Word count: 1433

Isoperimetric Quotients

Extracts from this document...

Introduction


Table of Contentsimage00.png

INTRODUCTION        

RIGHT ANGLE TRIANGLES        

Pair One.        

Pair Two        

Results        

ISOSCELES TRIANGLES        

Pair One        

Pair Two        

Results        

EQUILATERAL TRIANGLES        

Results        

FOUR SIDED SHAPES        

Squares        

Results        

Rectangles        

Results        

Irregular Four Sided Shapes        

Results        

FIVE-SIDED-SHAPES        

Pentagons        

Results        

Irregular Pentagons        

Results        

SIX-SIDED SHAPES        

Hexagons        

Results        

Irregular Hexagons        

Results        

GENERAL FORMULA        

Heptagon        

Octagon        

All regular shapes        

CIRCLE        

FINAL CONCLUSION        


Isoperimetric Quotients

Introduction

I am going to explore the different IQs for different shapes and try to find how the IQ relates to the shape. The formula for the IQ of a shape is

                                                IQ = 4∏xArea / Perimeter^2

        I will compare similar shapes, look at regular and irregular polygons and try to find patterns.

Right Angle Triangles

        I will start with Right Angle Triangles; I will look at four triangles with two pairs of similar triangles.

Pair One.

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        The two triangles I looked at had the same area but different IQs. This shows that IQs are not directly related to the area of the shape. In my next pair I will look at similar shaped triangles.

Pair Two

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The two triangles had identical IQs even though their area and perimeters were different. This shows that the IQ of a shape is related to its shape.

Results

I have found that IQs of the triangles I looked are not directly linked to their area, but to their shape.

...read more.

Middle

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As with the second pair of right-angled triangles, the similar shaped triangles had the same IQs.


Results

With both sets of triangles we can see that triangles that have identical perimeters or areas have different IQs, therefor the IQ of a shape is not linked to its area or perimeter. We also see that similar shapes have identical IQs. We also see that with a smaller difference between the length of the sides, the greater the IQ.

Equilateral Triangles

        I will now look at equilateral triangles, from the last two sets of triangles we can see that similar shapes produce identical IQs. Since all equilateral triangles are similar I predict that all equilateral triangles will have the same IQ.

        I will study three triangles to see if my theory is correct.


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Results

All of the IQs were the same, as I predicted. This is because all equilateral triangles are similar and IQs for similar shapes are identical.

Four Sided Shapes

I will now look at four sided shapes; I will look at squares, rectangles and irregular four sided shapes.

Squares

Since all squares are similar, like equilateral triangles, I expect all squares to have identical IQs.

        I will look at three squares to prove this.


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Results

As I predicted all the squares had identical IQs, also the square IQ was larger than the equilateral triangle IQ.

...read more.

Conclusion

         As you can see, the formula works for all regular shapes. We have seen that with an increasing number of sides of regular shapes, a larger IQ is given. The increase is exponentially decreasing e.g. triangle = 0.604599788

                                Square = 0.785398163

                                Increase of 0.180798975

                                Pentagon = 0.864806266

                                Increase of 0.079408103

Any series of numbers that display exponential properties can in theory continue indefinitely, but this series has a limit, being the number of sides. The shape with the largest number of sides is a circle, with an infinite number of sides. So far we have seen that all IQs are between 0 and 1. With a circle being the limit to the series, I expect the circle to have an IQ of 1.            

Circle

I will now investigate the circle, as I said I expect it to have an IQ of 1. I will also test my general regular formula on it.

        My prediction was correct; I will now test my regular shape formula.

        The circle works for my regular shape formula.

Final Conclusion

After reading through all my results and findings. I believe that the IQ of a shape is a measure of how compact/spherical it is. I think this because, with a smaller difference in sides, a larger IQ. Regular shapes have lager IQs than irregular shapes with the same number of sides. An increase in IQ is seen with larger numbers of sides and finally, a circle, the most compact/spherical shape possible, has the largest IQ possible.

...read more.

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