Triangle
I will now be looking at three different types of triangles – isosceles, right – angle and equilateral triangles. I will start off by looking at isosceles triangles to find the largest area with the perimeter of 1000m.
Isosceles triangles
400 ÷ 2 = 200 base x Height
H= 300² - 200² 2
300m 300m H= 90000- 40000 400 x 223.60
H= 50000 2
H= 50000 Area = 44720m
400m
300 ÷ 2 = 200 base x Height
H= 350² - 150² 2
350m 350m H= 122500- 22500 300 x 316.22
H= 100000 2
H= 100000 Area = 47433m
H= 316.22
300m
Isosceles triangles table of results
Equilateral triangle
333.3 ÷ 2 = 166.65
333.3m 333.3m H = 333.3² - 166.65²
H =111088.89 – 27772.22
H = 83316.67
H = 83316.67
H = 288.64
333.3m
Base x height
2
333.3 x 288.64
2
Area = 48101.85
Graph including equilateral triangle
Circle
I am now going to find out the area of a circle with a perimeter of 1000m to see if it gives the maximum area.
1000m
159.2m
The formula for the diameter is
C
D = R = D
2
1000
D = 318.5m R = 318.5
2
D = 318.5m
Radius = 159.2m
Pentagon
Now I am going to find the area of regular polygons. I will start off with a pentagon cause it has the least amount of sides in my tables and graphs
Pentagons have 5 sides and the perimeter is 1000m so each side must be 200 because 1000 ÷ 5 = 200
This is equivalent
To 5 triangles
Area of a triangle
200m
72 ÷ 2 = 36
200 ÷ 2 = 100
200m
36
H
100m
137.6
100m
Hexagon
A hexagon has 6 sides and has a perimeter of 1000m so I have to do 1000 ÷ 6 = 1666.66m
166.66m
30
H
83.333m
144.33
166.66
Heptagon
A heptagon has 7 sides and has a perimeter of 1000m so I have to do 1000 ÷ 7 = 142.857
142.857m
25.71
H
71.4285m
148.35
142.857
Octagon
An octagon has 8 sides and has a perimeter of 1000m so I have to do 1000 ÷ 8 = 125m
125m
22.5
H
62.5m
150.88
125
Nonagon
A nonagon has 9 sides and has a perimeter of 1000m so I have to do 1000 ÷ 9 = 111.11
111.11
20
H
55.555m
152.63
111.11
Decagon
A decagon has 10 sides and has a perimeter of 1000m so I have to do 1000 ÷ 10 = 100m
18
H
50m 50m
153.88
100m
Conclusion
After finding the area of the following shapes – rectangle, square, triangle, circle and the polygons, I found out that the shape with the maximum area is the circle. You can clearly see in the graph and table above that no other shape reaches the area of the circle, which makes it the shape with the maximum area. Looking back at the graph you can see that none of the other shapes even if you carry on will ever reach the area of the circle. So the shape that gives the biggest area, for the farmer to fence off his field is the circle.
By Joseph Marlow
