As you can see, the graph has formed a parabola. According to the table and the graph, the rectangle with a base of 250m has the greatest area. This shape is also called a square.
Now that I have found that a square has the greatest area of the rectangles group, I am going to find the triangle with the largest area. I am only going to use isosceles triangles because if I know the base I can work out the other 2 lengths because they are the same. If the base is 200m long then I can subtract that from 1000 and divide it by two. This means that I can say that:
Side = (1000 - 200) / 2 = 400
To work out the area I need to know the height of the triangle. Tow ork out the height I can use Pythagoras' Theorem. Below is the formula and area when using a base of 200m.
H² = h² - a²
H² = 400² - 100²
H² = 160000 - 10000
H² = 150000
H = 387.298
½ נ200 נ387.298 = 38729.833m
Below is a table of results for isosceles triangles from the base with 10m to a base with 500m.
Because the last two shapes have had the largest areas when they are regular, I am going to use regular shapes from now on.
The next shape that I am going to investigate is the pentagon.
Because there area 5 sides, I can divide it up into 5 segments. Each segment is an isosceles triangle with the top angle being 72º. This is because it is a fifth of 360º. This means that I can work out both the other angles by subtracting 72 from 180 and dividing the answer by 2. This gives 54º each. Because every isosceles triangle can be split into 2 right-angled triangles, I can work out the area of the triangle, using trigonometry. I also know that each side is 200m long, so the base of the triangle will be 100m.
Using SOH CAH TOA I can work out that I need to use Tangent.
H = 100 tan54 = 137.638
O = 100
T = tan 36
This gives me the length of H so I can work out the area.
Area = ½ נb נH = ½ נ100 נ137.638 = 6881.910
I now have the area of half of one of the segments, so I simply multiply that number by 10 and get the area of the shape.
Area = 6881.910 נ10 = 68819.096m²
All of the results that I have got so far have shown that as the number of sides increase, so to does the area. Using a spreadsheet and formula I have created a table that shows my prediction is right. This is show on the next page.
The formulae for the spreadsheet are:
To work out the base of a polygon you divide the perimeter of the polygon by the number of side (n)
To put this equation in to a spreadsheet, you must type the following:
=(1000/A3)
To work out the height of the triangle on a polygon, the equation is:
To put this equation in to a spreadsheet, you must type the following:
=(500/A3)/TAN(3.14/A3)
The equation to work out the area of the triangle is:
To put this equation in to a spreadsheet, you must type the following:
=(B3*C3)/2
To work out the area of the polygon, the equation is:
To put this equation in to a spreadsheet, you must type the following:
=(A3*D3)
From the method that I used to find the area for the pentagon I can work out a formula using N as the number of sides. To find the length of the base segment I would divide 1000 by the number of sides. Also on the next page is a graph showing the number of sides against area.
As you can see from the graph, the line straightens out as the number of side's increases. Because I am increasing the sides by large amounts and they are not changing I am going to see what the result is for a circle. Circles have an infinite number of sides, so I cannot find the area using the equation for the other shapes. I can find out the area by using π. To work out the circumference of the cir le the equation is πd. I can rearrange this so that diameter equals circumference/π. From that I can work out the area using the πr² equation.
DIAMETER = 1000 / π = 318.310
RADIUS = 318.310 / 2 = 159.155
AREA = π נ159.155² = 79577.472m²
From this I have concluded that a circle has the largest area when using a similar circumference. This means that the farmer should use a circle for her plot of land so that she can gain the maximum area.
