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I have been given the following equation to use to investigate Isoperimetric Quotients:-

Extracts from this document...

Introduction

I have been given the following equation to use to investigate Isoperimetric Quotients:-image00.png

I will begin by investigating the Isoperimetric Quotients of rectangles:-

Using this method I have gathered the following results:-

Width (cm)

Length (cm)

IQ

2

5

0.641*

3

24

0.310

12

17

0.762

2

10

0.436

3

7

0.659

1

4

0.503**

3

3

0.785

2

6

0.589***

1

3

0.589***

12

3

0.503**

4

6

0.754

6

15

0.641*

At this stage I can observe that IQ’s of rectangles are all less than 1. I

...read more.

Middle

image06.png

This proves that rectangles in the same ratio have the same IQ.

In the table the rectangle with the largest IQ was a Square, and the lowest IQ was a long, thin rectangle. I will now look further into this, and to keep it fair, both shapes will have the same area:-

This is another long, thin rectangle and has a low IQ.

This square has a high IQ. image08.png

I can conclude from this that the higher the difference between the lengths and widths of the rectangle, the lower the IQ will be.

Returning to the earlier statement, that rectangles in the same ratio have the same IQ, I can conclude that we can only work out a general formula for squares:-

Summary of rectangles:-

All rectangles in the same ratio have the same IQ. This means we can only find the pattern for regular shapes. The larger the difference

...read more.

Conclusion

I think this means that an Isoperimetric Quotient is a measurement of how efficiently a shape covers area, with a certain perimeter, or how it covers a certain area with as little perimeter as possible.

I will now investigate this, using a fixed perimeter of 80cm:-

image07.png

Summary of fixed perimeter investigation:-

The square has the smallest IQ, and also holds the smallest amount of area. The circle has the largest IQ and holds the most area.

CONCLUSION:-

The last investigation proves that IQ must be the measurement of how efficient a shape covers an area with a certain perimeter, or how little perimeter a shape will use to cover a certain area. The worst regular plane polygon is the triangle. There is a limit of 1 that only the circle reaches, and the number of sides a regular polygon has increases infinitely towards that limit. This is probably because as the number of sides increases, the shape becomes more efficient at containing an area.

...read more.

This student written piece of work is one of many that can be found in our GCSE Height and Weight of Pupils and other Mayfield High School investigations section.

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