# To investigate the areas of different shapes when they are joined together on square dotted paper

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Introduction

William Alston. G.C.S.E Maths Coursework. 1st October 2001 Introduction: - We are to investigate the areas of different shapes when they are joined together on square dotted paper. To start off with, in this investigation, I will be looking at regular shapes (those with 45� and 90� angles). To break it up into simple parts I will test 1 thing at a time using 5 different shapes. I will start by making regular shapes with 1 dot inside and then 2 dots and then 3 dots on the inside. Then I will make regular shapes and change the area each time e.g. 4, 5 and 6cm's�. Also I will try different numbers of dots on the outside if I feel it is necessary to my investigation. That is something that I will be deciding as I go along. Then I will look at irregular shapes (those with angles of say 30� or 75�). After I have drawn the 5 shapes I will put them in a table, so they are easy, to Asses. I will try to draw up a formula for those shapes then I will test the formula by drawing another shape and working out the formula before physically counting the area. Throughout the investigation I will be carrying out an on-going evaluation, as this will help me in finding formulas and noticing patterns. ...read more.

Middle

From testing different numbers of dots on the inside I have been able to draw up formulas that work and spot patterns between different numbers of dots on the inside and their formulas. There is definitely a link between the individual formulas but I am still unsuccessful in spotting that link. I have found formulas that work for individual numbers of dots on the inside but not yet one that works for any number of dots on the inside or in fact, any shape. I am now going to attack the problem by testing shapes with different areas. Maybe the formulas from that will point me more in the direction of finding the overall formula. Shapes with areas of 2 cm�: - Shape No. of dots inside No. of dots outside Area (cm�) 21 22 23 24 25 0 0 0 0 0 6 6 6 6 6 2 2 2 2 2 From the table we can see that there is a relationship between the number of dots on the outside and the area. The area is 1 less than half the number of dots on the outside. Therefore we can draw up the formula: - Testing the formula: - As you can see the formula works as when it was tested it was able to give me the area without counting it. ...read more.

Conclusion

I will now test the formula on a shape with irregular angles to make sure it can work for all shapes. In this shape there are 12 dots on the outside and 17 dots on the inside. I predict that the area of this shape will be: - After counting the shape by hand I can conclude that my prediction was right as it could work out the area of the shape. I believe that I have proved enough that this formula works for every single shape. The reason why the formula is dots outside divided by 2 is because, as I previously stated, as you increase the number of dots on the outside by one you increase the area by 0.5. Timesing by 0.5 is the same as dividing by 2. The reason that is +(dots in -1) is because 1 dot can touch the corners of up to 4 squares. So for every internal dot there is there can be 4 times as many squares. The reason why you take away 1 from the number of internal dots is shown below: - As I stated in my introduction I was planning on testing shapes with irregular angles such as 30� and 75�. I do not feel that it is necessary to test those as I have found and tested the formula that works for all shapes. I have also already proven that it works on shapes with irregular angles. ...read more.

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