I noticed from the graph the largest area was given by the rectangle, which is a square. So therefore there is no need to test this shape separately. The length and width of this shape is 250m by 250 m which is 62500
I then used my i.c.t skills to plot a graph between the relationship between width and area in order for me to find the maximum area within the rectangles. As you can see, the graph has formed a parabola.
Triangles
Isosceles triangle
Nest I drew a table for Isosceles triangle. I have kept 2 sides of the triangle the same in each triangle as 2 sides and 2 angles are equal in an isosceles.
The formula for a triangle is ½ x base x height
The formula used for column 3 which is a is =A4/2 Each time the coordinate changes.
The formula used for column 4 is =POWER(B4,2)-POWER(C4,2) Each time the coordinate changes.
We need to do this in order to work out the height.
The formula for column 5 which is height is =POWER(D4,0.5)
We need to work the height out in order to work out the area.
The formula for column 6 which is area is =(A4*E4)/2
I then used my i.c.t skills to plot a graph between the relationship between base and height in order to help me find the maximum area within an isosceles triangle.
I found from the graph the maximum area within a isosceles triangle is the triangle with the base 330m and the sides 335m each. I notice that this triangle had the least difference between the base and sides. It only had a difference of 5.
Equilateral Triangle
Next I tried equilateral triangle. As all the sides are the same length I could only investigate one equilateral triangle.
I draw a table with the base sides the same length because in an equilateral triangle all the sides are the same length
I noticed the only and largest area within equilateral is 4911252m
I could not plot a graph because I never had enough information for me to plot a graph.
I predict that regular shapes have the maximum area. I will now try regular pentagon, hexagon, heptagon, nonagon and decagon. I notice so far the number of sides a shape has the bigger the area.
Regular Polygons
I draw a table for regular polygons.
Area of Pentagon
Because there are 5 sides, I can divide it up into 5 segments. Each segments is an isosceles triangle, with the top angle being 72 degrees. This is because it is a fifth of 360. This means I can work out both the other angles by subtracting 27 from 180 (angles in a triangle add up to 180) and dividing the answer by 2. This gives 54 degrees each. Because every isosceles triangle can be split into equal 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 is 100m.
Using SOHCAHTOA I can work out that I need to use Tangent.
H= 137.683
O= 100
T tan 36
This has given me the length of H so I can work out the area.
This is a long way so I have found out a formula, which works for all regular polygons. This will save time.
To work out the base the formula I used was 1000/A3
adj x Tan θ = opp/adj x adj ( adjacent cancels out)
÷ Tan adj x Tan θ = opp/Tan θ
adj= opp/Tan θ
To work the height the formula I used was height = (1000/n)/2/Tan(360/n/2) I have simplified it below:
Numerator = 1000/n x ½ Denominator = 360/n x ½
= 100/2n = 360/2n
= 500/n = 180/n
Finale Formula
(500/n)/Tan(180/n))
The formula I used to calculate the area of one triangle was:
=(B2*C2)/2 = Base x Height/ 2 which is the formula to work out the area of one triangle. To work out the final area you multiply the area of one triangle by the number of sides of the polygon..
I have used my i.c.t skills to plot a graph to find the relationship between the number of sides in a polygon and the area. I noticed from my graph Decagon has the largest area as it has most sides.
My predication was right as the number of side’s increases so does the area. I also noticed from triangle, rectangle, square, pentagon, heptagon, octagon, nonagon and decagon the area is increasing and the shape is becoming more like a circle therefore I decided to try circle next.
Circle
The formula for the area of a circle is πrxr
We need to find r
We know circumference = 2πr
Circumference = Perimeter
We know the perimeter is = to 1000
Therefore 1000 = 2πr
To find out r you do r = 1000/2π
R= 159.1549431
Now we can work out the area.
Area = πrxr
Area = π x 159.159.1549431 x 159.1549.431
Area = 79577.47155m
The largest possible area for a circle is 79577.47144m squared .
Conclusion
I conclude that the area that which gave the farmer the most area is a circle. This is because it is a regular shape that has infinite number of sides, the maximum area that the farmer could get would be 79577.47155m squared. I also found out as the number of sides increases so does the area. Regular shapes give the largest area.
