Length x Width
Perimeter = 1000m ∴
If side A is 10m then side C is also 10m ∴
Sides B And D = 1000-20
Area = A x B (L x W)
10 x 490 = 4900m²
I decided to use excel spreadsheet as my way of storing my data, in excel it is easy to turn data into other sources or ways of presenting data. I decided that to add ten to side A every time and then see what the area works out to. Using excel is very easy it also saves time, instead of recording results manually and working them out you can insert formulas into the cells and the programme does the work for you.
Using this table I made a graph on excel of one width (A) against Area (AxB). This is on the next page.
The line of results on the graph has made a plane curve (parabola). From the table and the graph, we can see that the rectangle with a Length (A) of 250m has the greatest area. This shape is called a square and is the regular shape of rectangles.
Triangle
I then found the triangle with the largest area. I only used isosceles triangles because if I know the base I can work out the other 2 lengths because they are equal.
For example
If B = 200
1 Side = (1000 – 200) / 2 = 400
To work out the area I need to work out H (the height). To work out H (height) I can use Pythagoras’ Theorem. The formula and area a triangle with B (base) of 200m is shown below.
H² = h² - a²
H² = 400² - 100² ∴
H² = 160000 – 10000 ∴
H² = 150000 ∴
H = 387.298
½ × 200 × 387.298 = 38729.833m
The table and graph for the area of isosceles triangles with perimeter of 1000m is on the next page.
From the last two table of results I can see that regular shapes give the largest area so from now on I will only be investigating regular shapes.
The four sided shape has a larger area than the three sided shape therefore I can say that the more sides a shape has the larger the area will be, providing that you only compare the regular of each shape, so the next shape I investigated was the pentagon.
Pentagon
Every side is equal ∴
To work out what length each side is the formula is
1000 which equals 200
5
To work out the area you need to divide the pentagon up into 5 segments. Each segment is an isosceles triangle with the top angle being 72º because it is a fifth of 360º. You 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, from there you can work out the area of the triangle, using trigonometry. Each side is 200m long, so the base of the triangle is 100m.
Then using SOH CAH TOA (trigonometry) you can work out that you need to use Tangent.
H = 100 tan54 = 137.638
O = 100
T = tan 36
This gives 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
Now I have the area of half of one of the segments, all you need to do is multiply that number by 10 and get the area of the shape.
Area = 6881.910 × 10 = 68819.096m²
So from this I can be sure that the more sides a shape has the larger its area will be I can also be sure that the regular of each shape gives the largest area. Using the same method of working out as before I then worked out the area of a regular Hexagon and Heptagon.
I used the same method as before to work out the area of the 2 shapes.
Hexagon
1000 ÷ 6 = 166 1/6 ÷ 2 = 83 1/3.
360 ÷ 6 = 60 ÷ 2 = 30
Area = ½ * b * H = ½ * 83 1/3 * 144.338 = 6014.065
6014.065 * 12 = 72168.784m2
Heptagon
1000 ÷ 7 = 142.857 ÷ 2 = 71.429m
360 ÷ 7 = 51.429 ÷ 2 = 25.714 degrees
Area = ½ * b * H = ½ * 71.429 * 148.323 = 5297.260
5297.260 * 14 = 74161.644m2
