400 400 H² = h² - ½ b
H² = 400² - 100²
H² = √150000
H = 387.3
½ x 200 x 387.3 = 38730
A = 38730m²
I will now try an equilateral triangle otherwise known as a regular triangle seen as the last best shape was a regular shape:
H² = 333.3² - 166.7 ² = 83300m
H² = √83300
H = 288.6m
½ x 333.3 x 288.6
A = 48095.2m²
I will now go back to isosceles triangles to see if the area is bigger than that or an equilateral triangle:
260 260 H² = 260² - 240² = 1000m
H² = √1000
H = 100m
½ x 480 x 100 = 24000
240 A = 24000m²
480
Here are some more examples:
300 300 H² = 300² - 200² = 50000m
H² = √50000
H = 223.6m
½ x 400 x 223.6 = 44720
200 A = 44720m²
400
I have drawn a table so that I can see clearly which triangle had the biggest area:
As you can clearly see from my table, my results show that the equilateral triangle (or regular triangle) is able to obtain the largest area with a perimeter of 1000m.
Because of this pattern all shapes that Ii use now shall also be regular.
I will now investigate a regular pentagon; because there is five sides I can divide it into segments, each segment being an isosceles triangle with a top angle of 72° (one fifth of 360° is 72°). I can work out the other angles by subtracting 72 from 180 and then dividing the answer by two, that gives an answer of 54°.
Because every isosceles triangle can be split into 2 equal right angles triangles I can use trigonometry to work out the area of the triangle; I can then multiply the answer by 10 to give me the area of the full pentagon. I also know that each side is 200m long (1000 divided by 5), so the base of the triangle is 100m.
Each section of this regular pentagon will look like this:
Tan 36° = Opposite
Adjacent
0.73 = 100
?
100 / tan 36° = 137.6
Area of a triangle = ½ base x height
200 / 2 = 100
100 x 137.6 = 13,760m²
13,760 x 5 = 68,800m²
Area of a pentagon in terms of n:
360 360 n 2n
1000 1000
n n 2n
1000 / Tan 360
2n 2n
1000 x 1000 / tan 360
2n 2n 2n
1000 x 1000 / tan 360 x n
2n 2n 2n
So far I have found that as the number of sides of the shape increases so does the area, I will now a regular hexagon and a regular heptagon, I will use the same method as I used to work out the area of the pentagon for these next two shapes.
1000 / 6 = 166 / 2 = 83
360 / 6 = 60 / 2 = 30
Cos 60° = Adjacent
Hypotenuse
0.5 = 83
?
= 166.6666667
166.6666667² - 83.3333335²
= √20833.33332
h = 144.3
Area = ½ x b x h
= ½ x 83 x 144.3
= 6012.5
6012.5 x 12 = 72150
A = 72150m²
Heptagon
1000 / 7 = 142.9
142.9 / 2 = 71.5
360 / 7 = 51.4
51.4 / 2 = 25.7
Tan = Opposite
Adjacent
B = 71.4
h = 148.3
A = ½ x b x h
= ½ x 71.4 x 148.3
= 5294.3
5294.3 x 14
= 74120.2
A = 74120.2m²
Octagon
It is made up of 8 separate isosceles triangles, each looking like this:
Tan = Opposite
Adjacent
Tan 22.5º = 0.41
0.41 = 62.5
?
62.5 / Tan 22.5º – 150.89
Area = 125 / 2 =62.5
125 x 150.89 = 9430.625m²
= 9430.625 x 8 = 75,445
A = 75445 m²
The area of the regular octagon is larger than the area of the regular pentagon and the square. I have decided to try a regular decagon as my next shape to investigate. If am right, the pattern should carry on, and the regular decagon should have a larger area than the pervious shapes I have investigated.
Decagon
Each of the ten isosceles triangles that make up this regular decagon look like this:
Tan = Opposite
Adjacent
Tan 18º = 0.32
0.32 = 50
?
50 / Tan 18º = 153.88
100 / 2 = 50
50 x 153.88 = 7,694
7,694 x 10 = 76,940
A = 76,940m²
1000 / Tan 360
2 x 12 2 x 12
41.6 / 0.2679
= 155.2819
1000 x 1000 Tan 360
2 x 12 2 x 12 2 x 12
41.6 x 41.6 / Tan 15
= 6,459.7238
1000 x 1000 Tan 360 x 12
2 x 12 2 x 12 2 x 12
41.6 x 41.6 / Tan 15 x 12
= 77,516.6853
A = 77,516.6853 m²
In order to show more clearly how the area is increasing with the number of sides I have drawn up a table to show sides against area:
As I have already come to the conclusion that as the number of side’s increases the area increases I am going to work out the area of a circle. Circles have infinite sides and should prove my conclusion to be correct.
*r = radius*
c = 1000 = 2πr
500 = πr
Area of a circle = πr²
πr² = 79,622m²
500 = π x Ans² = 79,577.47155
π
A = 79,577.5m2
From this I conclude that a circle has the largest area when given a perimeter of 1000m.
