Triangles
The next kind of shape I am going to investigate will be the triangle.
If the base is ‘B’ then we can work out the length of the two other sides using the following formula:
Length of x = (1000- B) / 2
Now that we know the length of the sides we can then work out the height which we will need to find the area of the triangle. To do this we use Pythagoras.
Pythagoras’s Theorem is Z² = X² + Y²
Having found the base (B) and the side (X) we then split the triangle in half giving us a right angled triangle. The height (H) then is worked out by this formula:-
H² = X² - ½ B²
This formula then gives us the height and from here we can work out the area with the formula:-
Area of Triangle = ½ Base x Height.
Below I have put all the data into a table so that I can work out the areas in a quicker time. I will increase the length of the base in lots of 10. To make the table easier to read I rounded up the figures to the nearest whole number.
From the table above it is clear to see that when the base is 330 meter then the maximum area is obtained in 48105m².
To check all possibilities I then drew up another table for the figures in-between 332 and 337. This would give me a true reading as to which triangle would give me the greatest area.
From this more detailed table it is clear that if she was to have a base of 333 meters then she would gain maximum area in 48122.5 meters².
From this I can see that a regular shape will have a greater area than an irregular form of a certain shape. Then next idea I would like to put forward is that all polygons are made up of a number of isosceles triangles. I can show this idea by investigating the area of a pentagon.
Pentagon
From the diagram we can see that we can divide the pentagon into 5 triangles which is also the number of sides it has. By working out the area of one of these triangles we can find the area for the whole pentagon.
To work out the area of theses triangles I need to use trigonometry and to use the sine, tangent, and cosine rules, I need as much data as possible.
I can firstly find the length of the base by dividing 1000 by the number of sides (n). So : -
1000 meters / 5 = 200 meters
Next to find the angle (a) I divide 360 degrees by the number of sides. So:-
360 degrees / 5 = 72 degrees
By having this we can then find the other two angles by taking 72 degrees from 180 and dividing the answer by two as the other two angles in the triangle will be equal.
(180 degrees – 72 degrees) / 2
The next length you need to find is the length of the height of the triangles. To do this you use trigonometry. To make it easier we divide the triangle into 2 down the middle. So we are given the shape and data below.
I will need to use tangent because:
Tangent = opposite / adjacent
So the equation will be :
Tan 54 = Height / 100
Height = Tan 54 x 100
= 137.638192m
So now we use the formula
Area of triangle = (½ Base) x Height
Giving an area of:
(200 / 2) x 137.638192m
= 13763.8192 meters ² and all that is needed to multiply this figure
by 5 to give the final area of the pentagon.
13763.8192 meters ² x 5
Area of Pentagon = 68819.096 meters ²
Having area of a pentagon, a five sided formula I can use the workings above to create a formula that will give me an area of an n-sided polygon.
N-Sided Polygon
Below I have created a formula to work out the area of an n-sided polygon and I have annotated it to show how I have come to it.
N (500/n x [ (500/n) / tan (180 / n) ] )
As you can see from the formula above in blue I have highlighted the length of ½ base and in yellow I have highlighted the height of ach isosceles triangle. By multiplying the two together we can get the area of each isosceles triangle in the shape. The ‘N’ at the beginning will give the total area of the regular polygon by multiplying the area of
each triangle by the number of triangles.
For this formula to work all that you have to do is to replace the letter n with the number of sides for each polygon. This makes my investigation a lot easier as I can investigate shapes with millions of sides. To start with I worked out a 5 sided polygon to check my formula worked. As you can see on the following page I got very accurate results.
N.B.
From the graph above you can see that I have had to go into many decimal places because other wise you would not be able to see the small differences between the larger numbers. The is obviously an upward trend and this would continue into much higher numbers such as the trillions however for this coursework my computer would not be ale to handle those kind of numbers. From my results however that I have taken a billion sided polygon has the highest of all the polygons I have calculated for.
Circle
The final shape that I will investigate in my coursework is a circle. I predict that this will have the largest area because of my previous calculations. I calculated that as the number of sides on a polygon increased then so would the area. With a circle there are an unlimited number of sides meaning it will have the biggest area.
As we have the perimeter (or circumference) of the circle to start with I will need to work back wards to find the radius then I can work out the area. My calculations begin on the following page.
Circumference = 2 x π x radius
So
Radius = 1000 / 2 π
= 159.1549431
and then to find the area we use the formula:
Area of a Circle = π x radius ²
= 159.1549431 ² x π
= 79577.47155 m²
So above is the area for a circle and that is the final shape I will work out. I will now put the areas of the different shapes I have investigated into a table to compare them.
Results
