All of these results fit into the graph line that I have, making my graph reliable.
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. Because in any scalene triangle, there is more than 1 variable, there are countless combinations, so I am only going to use isosceles triangles. This is because if know the base length, then 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 2. This means that I can say that:
Side = (1000 – 200) ÷ 2 = 400
Because the regular rectangle was the largest before, I added 333.3 as a base length. This is the length of the base of a regular triangle. It is also an equilateral triangle.
The regular triangle seems to have the largest area out of all the areas but to make sure I am going to find out the area for values just around 333.
This has proved that once again, the regular shape has the largest area.
Because the last 2 shapes have had the largest areas when they are regular, I am going to use regular shapes from now on. This would also be a lot easier as many of the other shapes have millions of different variables.
I will now split the triangle in half, giving me an angle of 36° at the tip of the triangle.
Using SOHCAHTOA I can work out that I need to use Tangent.
=137.638
This has given me the length of H so I can work out the area.
Area = ½ X b X H = ½ x 100 X 137.638 = 6881.910
I now have the area of half of one of the 5 segments, so I simply multiply that number by 10 and I get the area of the shape
Area = 6881.910 X 10 = 68819.096m2
All of the results that I have got so far have shown that as the number of side’s increases, the area increases. I am going to investigate this further with a regular hexagon (6 sides) and a regular heptagon (7 sides).
I am going to work out the area of the 2 shapes using the same method as before.
Hexagon:
1000 ÷ 6 = 166 1/6 ÷ 2 = 83 1/3.
360 ÷ 6 = 60 ÷ 2 = 30
Area = ½ X b X H = ½ x 83 1/3 X 144.338 = 6014.065
6014.065 X 12 = 72168.784m2
Heptagon:
1000 ÷ 7 = 142.857 ÷ 2 = 71.429
360 ÷ 7 = 51.429 ÷ 2 = 25.714
Area = ½ X b X H = ½ X 71.429 X 148.323 = 5297.260
5297.260 X 14 = 74161.644m2
My predictions were correct and as the number of side’s increases, the area increases. Below is a table of the number of sides against area
From the method that I used to find the area for the pentagon, hexagon and heptagon I can work out a formula using n as the number of sides. To find the length of the base of a segment I would divide 1000 by the number of sides, so I could put , but as I need to find half of that value I need to put . All
The method that I used above has been put into an equation below.
Next, I am going to look at the Octagon. The octagon is a regular 8-sided shape, made up of 8 isosceles triangles.
To find the length of a side, you need to divide 1000 (the perimeter due to the amount of fence available) by 8, the number of sides an octagon has.
1000m/ 8 = 125m
I will now look at one of the 8 segments to find out its area. The angle at the top of the triangle, round the outside, we know is 360°. The find the angle on the inside, we have to divide 360° by the number of sides the pentagon has.
360/ 8 = 45
We know that the interior angle of the whole triangle is 45°. We need to split the triangle in half though so we can use trigonometry. This means that the angle halves, making it 22.5°.
We already know that the base of the triangle is a side, therefore making it 125, but as we are concentrating on half the triangle, that number has to be halved also, giving us 62.5.
Using SOCAHTOA, we can see that the two sides involved are the opposite and the adjacent, meaning we need to use tangent.
Tan22.5° = O/A
= 62.5 / h*
h = 62.5/ Tan22.5°
= 150.8883476 m2
* The ‘h’ is the height I am trying to find, not the hypotenuse
Now I can find the area of the whole triangle.
Area = ½ X b X H
Area = ½ X 125 X 150.8883476 m2
Area = 9430.521725 m2
I can now multiply the answer by 8, as I have the area for one of the 8 segments in an octagon.
Area of a octagon = Area of 1 segment X 8
Area of a octagon = 9430.521725 m2 X 8
Area of a octagon = 75444.1738 m2
You can notice from the findings of areas from the previous shapes that as the number of sides increase, as does the area.
There is a formula you can use instead of drawing and working out each shape. It is as follows:
[500/n X (500/n) / Tan (180/n)] X n
That is the full equation and it works on all of the shapes that I have already done, giving the same answers as before. Below is a table showing the answers I got when I used the equation.
Now that I have this equation, I am going to use it to work out the area for a regular nonagon and decagon.
As you can see, the larger the number sides on the shape, the larger the area is. This pattern has carried on going for all of the shapes that I have investigated, so I am going to investigate shapes with the following amount of sides:
20, 50, 100, 200, 500, 1000
On the following page is a table showing the results that I got.
The pattern remains the same as the number of sides goes up. You can clearly see that as the number of sides increases, the area does to. This is the same from a three- sided shape, all the way up to 1000. If this is the case, then what would the area be of a shape that at an infinite number of sides?
The final shape I am going to look at is the circle. Circles have an infinite amount of sides, so I cannot find the area by using the equation that I have used to find the other amount of sides out. I can find out the area of a circle by using π r2. But we don’t know the radius yet, so we have to work out the diameter. The circumference of a circle can be found using the equation is π d. I can rearrange this so that the diameter is circumference/ π. From that I can work out the area using the π r2 equation.
We know that the circumference has to be 1000, as that’s the amount of fencing given.
1000/ π = 318.310
Now we have the diameter of the circle. To find the radius, we simply divide by two, as the radius is half the diameter.
318.310/2 = 159.155
So we know have the radius. To find the area of a circle, you multiply π, by the radius squared.
π X 159.1552 = 79577.472m2
