All of these results fit into the graph line that I have, making my graph reliable. I have solved that a square has the greatest area of the rectangles group.
Triangles:
My next task is to find the triangle with the largest area. In any scalene or three angled triangle, there is more than one variable so therefore there are "n" combinations. So I'm only going to use isosceles triangles for now. This is due to the fact that if I know the base length, then bearing in mind that the perimeter has to add up to 1000m and I already have the base length I can work out the other two sides because they are parallel. If the base is 200m long then I can subtract that from 1000 and divide it by 2. This can be rearranged to the following formula:
Side = (1000 - 200) ?= 400
I have used this formula to work out the area when the base is at different heights. To do this I need to use Pythagoras' theorem. Below is a diagram of an isosceles triangle. Again, isoceles triangles, like squares, are good foundations to start
H is the height of the dotted line. B is the length of the base, A is half the base length, and h is the hypotenuse of the right angled triangle.
To work out the area I need to know the height of the triangle, the heigh of H- the dotted line. To calculate this I can employ Pythagoras’s theorem, and below is the formula and area when using a base of 200 metres.
H² = h²-a²
H²= 400²-100²
H² = 150000
H= 387.298
0.5 X 200 X 387.298 = 38729.833 m
Below is a table of results for isoceles triangles, and the variable here is the base in metres, as I change it every 50 metres until I get closer to pin-pointing the base which brings the greatest area.
My results tell me that the regular triangle has the largest area compared to all the areas. To back up my evidence I am going to find out the area for values just around 333, as with rectangles.
This has proved that once again, the regular shape has the largest area.
Greater Detail on Triangles.
Base = 333
H2 = 3332 - 166.752
H2 = 110889 - 27805.5625
H = 83083.4375
H = 288.2419773
A = ? x b x h
A = ? x 333 x 288.2419773
A = 47992.28922m2
A = 47992.29m2
Base = 333 1/3
H2 = 333 1/32 - 166.66672
H2 = 111111.1111 - 27777.78889
H = 83333.33
H = 288.675
A = ? x b x h
A = ? x 333 1/3 x 288.675
A = 48112.52243m2
A = 48112.5m2
Base = 333.5
H2 = 333.52 - 166.6252
H2 = 111222.25 - 27763.89063
H = 83458.35937
H = 288.8916049
A = ? x b x h
A = ? x 333.5 x 288.8916049
A = 48172.67512m2
A = 48172.68m2
Base = 334
H2 = 3342 - 166.52
H2 = 111556 - 27722.25
H = 83833.75
H = 289.5405844
A = ? x b x h
A = ? x 334 x 289.5405844
A = 48353.27759m2
A = 48353.278m2
Now that I have gone into detail on rectangles, triangles and rhombus's, I can extend my investigation by finding 5-sided shapes, 10-sided shapes and 100-sided shapes. My first investigation will be on a regular pentagon. Once I have investigated these polygons, I can hopefully find a formula for n number of side.
Pentagons:
Due to the last 2 shapes have had the largest areas when they are regular, I am going to use regular shapes. This would also be a lot easier as many of the other shapes have different variables. So now we can investigate regular shapes e.g. 5-sided shapes, 10-sided shapes, 100-sided shapes etc.
The next shape that I am going to investigate is the pentagon.
Using Trigonometry, I can work out that I need to use Tangent (T)
O 100
T tan36
This has given me the length of H so I can work out the area. (X) is a multiplier.
Area = ? X b X H = ? x 100 X 137.638 = 6881.9 m2
I now have the area of half of one of the 5 segments, so I simply multiply that number by 10 and I get the area of the shape:
Area = 6881.9 X 10 = 68819m2.
All of the results that I have got so far have shown that as the number of side's increases, the area increases. I am going to investigate this further with a regular hexagon (6 sides) and a regular heptagon (7 sides) going up to 1000000-sided shape.
am going to work out the area of the 14 shapes using the same method as before.
Hexagon:
1000/6 = 166 2/3
360/6 = 60
180 - 60 = 120
120/2 = 60
Tan 60 x 831/3 x 831/3 x 6
= 72169m2
Decagon: 10-sided shape.
1000/10 = 100
360/10 = 36
180 - 36 = 144
144/2 = 72
Tan 72 x 50 x 50 x 10
= 76942.08843 m2
100-sided shapes.
1000/100 = 10
360/100 = 3.6
180 - 3.6 = 176.4
176.4/2 = 88.2
Tan 88.2 x 5 x 5 x 100
= 79551.28988 m2
You can clearly see, that as the number of sides in a shape increase, the area within them increase.
My predictions were correct and as the number of side's increases, the area increases. Below is a table of the number of sides against area:
From the method that I have used to find the area for the pentagon, hexagon and heptagon I can work out a formula using n as the number of sides. To find the length of the base of a segment I would divide 1000 by the number of sides, so I could put , but as I need to find half of that value I need to put .
The method that I used above has been put into an equation below.
1000 ?= 142.857 ?=
360 ?= 51.429 ?=
Area = ? X b X H = ? X 71.429 X 148.323 = X
5297.260 X 14 =
Above is the full equation and it works on all of the shapes that I have already done, giving the same answers. Below is a table showing the results I got when I used the equation (above).
Now that I have this equation, I am going to use it to work out the area for a regular octagon, nonagon and decagon.
As you can see, the larger the number sides on the shape, the larger the area is. This is because, with the addittion of an extra side, you are making the shape more concvave, and thus the area increaes.This pattern has carried on going for all of the shapes that I have investigated, so I am going to investigate shapes with the following amount of sides:
20, 50, 100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000 and 100000.
Below is a table showing the results that I achieved.
We can clearly see, that as you pass 20,000 sides, the total area of the polygon, becomes the same, and this would be proved by a straight platuea in a graph.
Although I have simply tested regular polygons, we know, because of their concave nature, they will always provide the greatest possible area. An irregular hexagon for instance, because of its convex nature, would always offer a smaller area. From this knowledge we an discount irregular polygons as being contenders for the greatest area. Also, since we have inspected squares and triangles originally, we are not prepared to widen our scope to irregular shapes.
Circles:
I am now going to see what the result is for a circle, because the larger the number of sides there are the more the regular shape is going to look like a circle. I already know that circles have an infinite amount of sides and degrees, so I cannot find the area by using the equation that I have used to find the other amount of sides out. I can find out the area of a circle by using ( ). To work out the circumference of a circle the equation is ( ) d. I can rearrange this so that the diameter is circumference/( ). From that I can work out the area using the ( ) r2 equation:
1000/( ) = 318.30988618379067153776752674503
= 318.30988618379067153776752674503 /2
= 159.15494309189533576888376337251
( ) x 159.154943091895335768883763372512
= 79577.47154594766788444188168625m2
As the number of sides in a regular polygon increase towards infinity, we achieve a greater area, and it is logical that a circle, which is defined as having an infinite number of sides, will achieve the greatest. Its curved nature means that there is no part which is convex, and hinder the area within. From this I conclude that a circle has the largest area when using a similar circumference.
