# The aim of this investigation is to find what is the maximum area you can obtain with the perimeter of 1000m.

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Introduction

Nisha Patel 10E

Mr.McDonagh

The aim of this investigation is to find what is the maximum area you can obtain with the perimeter of 1000m.

To achieve this solution, I’ll have to work through it logically. To do this I have decided to start investigating different types of triangles seeing as this is the only possible shape to make with the least number of sides. I will use Hero’s formula to fin the area of the triangles.

_______________________________

Hero’s Formula: sqrt /500 (500 – A) (500 – B) (500 –C)

B NOT POSSIBLE TO DRAW

A C

B

A C

B

A C

B

A C

B

A C

B

A C

Whilst drawing these triangles you can’t have triangles with any of the lengths of 500m or more because if one side is 500m the other two have to add up to 500 because the maximum perimeter is 1000m. If you did draw this shape it would look like this:

A B C

Basically it is a straight line.

As you can see how different triangles with the same perimeter have different areas. Below is a table with different areas from the triangle above. I have put it in a table because it is easier to analyze and evaluate.

The formula that I entered

The formula used here is:

=SQRT (500*(500-A2)*(500-B2)

Middle

so, Area of triangle = 100 x 400.9772

= 46097.72m^2

I will use his method with the rest of the values. Below is a table showing the results of the values.

Now we can transform this into a graph, which is easier to read.

This graph is a parabola. It is symmetrical. The highest point of the graph is where the area is the largest. We cannot estimate the highest point but you can tell that it is between the values 300 and 350. I can use these values as margins for when I home in to find the real answer.

The triangle with the largest area has the lengths of 333.3, 333.3, and 333.3, which is a regular equilateral triangle.

Now all I have to do is find the area of this triangle and then I’m done.

Area of Triangle= sqrt (500(500-333.3)(500-

333.3)(500-333.3))

=SQRT 500 x 166.67 x166.67 x

166.67

=sqrt 2314953706.4815

I have done the three sided shape now I have to do the four sided, which is the parallelogram

Now I am going to investigate parallelograms because this is the next shape that has four sides after the three-sided triangles. In this section I have to find out the parallelogram with the perimeter of 1000 metres with the largest are just like the triangles.

E.g.

h

Conclusion

The relationship in this graph shows that as the number of sides increases so does the area.

A graph to show the elation between the number of sides and the area

Now my aim is to find a formula, which can be used to find out the area of the polygon.

n= number of sides .

What I have to find out:

FORMULA= xyn

x

y

x= 500

n__

tan(180)

n

y= 5oo

n

500 5oo

n___ x nn

tan180

n

. We could try this formula to see if it works by applying it to one of the polygons that we have already done. I am going to try the heptagon.

500^2

n tan(180)

n

Area= 250000

tan(180/7)

Area= 250000 x n

tan 25

Area= 250000 x n

3.264

Area= 74162.27584m^2

I can also use this formula to find the area of any other polygon with any sides. Although you cannot use it to find the area of a circle because we do not know how many sides it has because it is an infinite number. This is the answer to this investigation. The circle is the shape which has the largest area inside because it as an infinite number of sides.

To find the area you would have to use πr^2. We do not know what r is but you could find out by re- arranging the formula for the circumference.

1000= 2πr

500= πr

r=159.1549

∴π x 159.1549 x 15901549 = area of a circle

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

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