1000 = 333.3m.
3
I need to keep in mind that the answer is
actually 333.3 recurring.
333.3m
(a) To work out the area I need to find the height of the
triangle. To do this I will split it into 2 right-angled
triangles and use Pythagoras theorem.
166.6m
a = 333.3² - 166.6²
a = 111111.1 – 27777.7
a = √83333.4111
a = 288.7m
To find the area of a triangle you need to multiply ½base x height.
Therefore area = 0.5 x 333.3 x 288.7
Area = 48116.7m
I will now investigate isosceles triangles
Firstly I will take a measurement for the base. As the other two sides have the same lengths, to obtain them I can subtract the base from 1000. Then to get one side I can divide the answer by two.
To obtain the area you need to again use Pythagoras. When investigating the triangles I discovered that the base must be below 500 otherwise it will not be correct due to the fact that the other sides would not reach a sufficient point.
The following table shows the formulas I used to obtain the results for the different isosceles triangles.
Here are my results:
The isosceles triangle with the largest area is the triangle with a base of 350m and each side of 325m.
Because the last two shapes have had the largest areas when they are regular, I am going to use regular shapes from now on.
I am now going to investigate the pentagon.
Because there are 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 triangles will be 200m.
72º
200m
54º 54º
200m
Use TAN
H = tan54º
H 100
H = 100 x tan54º
H = 137.638m
54º
100m
From this I can now calculate the area of the pentagon segment.
½ x base x height
50m x 137.638m = 6881.9m²
I now have the area of half of one of the segments, so I now multiply that number by 10 and get the area of the shape.
Area of pentagon = 6881.9 x 10 = 68819m²
All of the results that I have got so far have shown that as the number of sides increase, so to does the area. Therefore, using formulas, I am going to try to calculate the areas of other polygons.
The spreadsheet formulas and the results are on the next page.
I have discovered that the areas of regular polygons continue to increase but only very slightly. An immense number such as 1 million would be either very impractical or may probably be impossible.
I am now going to investigate the 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 pi. To work out the circumference of the circle the equation is π x diameter. I can rearrange this so that diameter equals the 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 × 159.155² = 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.
