n = length of side.
As you can see from the above table, the value that produces the highest area is 250. This would make the sides of the 4 sided shapes 250 by 250, which is a square.
This is the shape that gives the highest area for four sided shape.
3 Sided Shapes
Once I had worked out the formula for squares I decided to research triangles. I started off with an equilateral triangle.
I knew that the area of a triangle was
THE SQUARE ROOT OF: s(s-a)(s-b)(s-c)
Using this formula I could work out the area of an equilateral triangle shaped fence.
* = Multiplied
THE SQUARE ROOT OF (500*(500-333.333…)*(500-333.333…)*(500-333.333…)) = 48112.52245 m
I then tried this formula with an isosceles triangle.
THE SQUARE ROOT OF (500*(500-400)*(500-400)*(500-200)) =
38729.83346m
When I had worked out this formula I discovered that the two areas that had given the largest areas for 3 and 4 sided shapes had been regular shapes. I wondered if this meant that regular shapes gave the largest areas. I thought that it might have something to do with the lines of symmetry in a shape.
5 Sided Shapes
Once I had worked out the largest areas of the squares and triangles I realised that a shape with the most lines of symmetry had the largest area. This meant that regular polygons would give the largest area, so from now on I will only look at regular shapes.
I then began to study pentagons. A pentagon is a five sided shape. To work out a general formula for this coursework I realised that I needed a formula that could link up all shapes. I was only looking at regular shapes, and I realised that when you looked at the lines of symmetry and drew lines from the vertices to the centre of the shape Isosceles triangles were formed. The angles inside are 360 when n is the number of sides.
This means that the area of any regular shape can be found by finding the area of the isosceles triangle and multiplying by n (the number of sides).
To work out the area of the triangle I have to divide it into two.
Tan (180) is equal to 500 divided by the height of the triangle created by the shape.
h = height
So Tan (180) = 500 : n
n h
So h = 500 : n
Tan (180)
n
This means that the area of the isosceles triangle is
1 * 1000 * 500 : n
2 n Tan (180)
I have added in the 1 and the 1000 because the area of a triangle is half of
2 n
the base times the height.
This means that the general formula for any regular sided shape is
n (250000 : Tan (180))
n n
OR
250000 : Tan (180)
n n
With this formula I can work out the areas of all regular sided shapes.
The area for a regular pentagon is therefore 68819.09602m
I then began to look at hexagons, which are six sided shapes
6 Sided Shapes
I then began to look at hexagons, which are six sided shapes
Using my formula I worked out that the area of the regular hexagon was 72168.78365m.
7 Sided Shapes
I then looked at heptagons, which are 7 sided shapes.
The area for a regular heptagon was 74161.47845m
8 Sided Shapes
I then looked at octagons, which are 8 sided shapes.
The area for a regular octagon was 75444.17382m.
Finding the Formula
When I had worked out these areas I made a table of my results and predicted some areas using my formula
n = Number of sides
I then made a graph of this table.
Looking over my graph made me realise that the higher the number of sides a polygon had the larger its area was. This would mean that a circle would have the biggest area, as it has infinite sides. If this was the case then I wondered how my formula would fit a circle. My first step was realising that as n got towards infinity, that tan (180) ≈ 180.
I can demonstrate this best in a table.
n= Number of sides
n is an infinite number.
You can see the numbers get closer together, and they will eventually meet as n nears infinity
This meant that my general formula was now 250000 : 180
n n
However, the formula for a circle involves multiplication, so I had to make my formula
250000 * n
n 180
I cAN now cancel out the two n’s, leaving me with the formula
250000 * 180
Then I realised that 250000 was my original r , as my radius was originally 500. This meant that my formula was now
r * 180
My formula was now beginning to resemble the formula for a circle, but I still had to convert the 180 into π.
I realised that it must involve radians.
Radians
A radian is the angle subtended at the centre of the circle by an arc of length equal to the radius.
In radians 360° is equal to 2π. This means that 180° is equal to π, as 180 is half of 360 and π is half of 2π.
I now had my formula
r * π
Or, in its familiar form
πr , which is the formula for a circle..
In my investigation I discovered that a circle would give the largest area for a fence of 1000m perimeter as it has an infinite number of sides.
I also discovered that the general formula for any n sided polygon is
250000: tan (180)
n n
