The perimeter=1000meters because 450 + 450 + 100 the area= 22360.67978meters²
Because: -
C²=A²+B²
450²=50²+B²
202500=2500+B²
200000=B²
√200000=447.2135955
Height=447.2135955
The perimeter is 1000meter and the area= 75.62368 meters² because: -
C²=A²+B²
275²=225²+B²
75625=50625+B²
75625-50625=25000
√25000=158.113883
Height=158.113883
The perimeter is=1000 because 475+475+50 the area= 11858.54123meters²
Because: -
Equilateral triangle
The perimeter is 1000meter and the area=48112.52243 meters²because: -
Conclusion
I found that the triangle that had the same sized lengths had the biggest area. The wider triangle had a large area compared to the higher triangle. I found that the triangle had the same theory as the squares, the near you get to a perfect shape (all the sides are the same) the large the area.
Investigating a Pentagon
Aim
I have to decided to find out the area of shapes with more sides to see if the area will increase compared with the square and triangle. For this irregular pentagon I am first going to divide the square and triangle up and work out the area of them then add them together to get the area.
Irregular pentagon
The perimeter is 1000 meters because 250 + 250 + 100 + 100 + 300=1000
The area=45000 meters²
Because: -
I split the pentagon in to a triangle and a square then I found the area of the two and add them up: -
Square: -
(B*H)=area
100*300=30000
Square + triangle = area
30000 + 15000= 45000
Regular Pentagon
For this sided shape and the other shapes like this I am going to first divide the pentagon in to triangles. Then work out the area of the triangle and then times by the number triangles I have split.
The perimeter of the pentagon is 1000 the area= 68819.09602meters² because
I first divided the pentagon in to 5 triangles then worked out the angles of the triangle. Then I worked out the height (o) Tan54=O/100
Tan54=1.37638192*100=137.638192
(HB)/2=13763.8192
13763.8192*5=68819.09602
Conclusion
I found that the first pentagon had a smaller area compared to the second. I think the second pentagon has a bigger area because all the sides are equal.
First pentagon=45000
Second pentagon=68819.09602
Investigating a hexagon
Irregular hexagon
The perimeter is 1000meter and the area= 36806.07966meters² because
I first divided the hexagon into triangles then I got one of the triangles (200m base) then I worked out the area of that then I times it by 2. Then I worked out the area of the other triangle (150m base) and times it by 4.
The area of the first triangle=
Tan60=O/100
Tan60=1.732050808*100=173.2050808
(HB)/2=173.2050808*100/2=
8660.254038
The first triangle*2=17320.50808
The second triangle*4=19485.557158+17320.50808=36806.07966
Regular hexagon
The perimeter is 1000meter because 60 + 60 + 60 + 60 + 60 + 60=1000
The area= 72188.78365meters² because
First I would divide the hexagon into 6 triangles. Then I found out what the area is and times the area by 6 to get the area the hexagon.
Tan60=1.732050808
Tan60=O/83⅓
Tan60*83⅓=144.3375673
Height=144.3375673
Conclusion
I found that the regular the shape the large the area so I am going to stop doing irregular shapes because I know that the regular are going to have a large surface area.
Investigating a Octagon
The perimeter is 1000 because 125*8=1000
The area= 75444.17382meters²
Because
Tan67.5=2.414213562
Tan67.5=O/62.5
Tan67.5*62.5
2.414213562*62.5=
150.8883476
Height=150.8883476
Conclusion
I found that the area for the octagon had a large surface area compared with the hexagon this is because the octagon is nearer to the perfect shape.
Investigating a Decagon
The perimeter =1000 because 100*10=1000
The area =76942.08843meters²
Because
Tan72=3.077683537
Tan72=O/50
Tan72*50=153.8841769
Conclusion
I found that the more sides you have the larger the area so I have decided to look at shapes with more sides such as a 20,50,100 sided shape then a circle.
Investigating a 20 sides shape
The perimeter of the shape is 1000 because 1000/20=50 and there are 50 sides
The area=78921.89393meter²
Because
Tan81=6.313751515
Tan81=O/25
Tan81*25=157.8437879
Height=157.8437879
The 20-sided regular shape I have just done Has a larger area compared with the 10 decagon.
Investigating a 50 sided shape
The perimeter 1000meters because 20*50=1000
The area=30514.10338meters²
Because
Tan86.4=15.89454484
Tan=O/A
15.25705169=O/10
15.25705169*10=
152.5705169
Height=158.9454484
Conclusion
I am finding that the area is increasing when have more sides so I am now going the do a 100-sided shape and then a circle. to test my idea
Investigating a 100 sided shape
The perimeter is 1000 because 10*100=1000
The area=79551.28988meters²
Because
Tan88.2=31.82051595
Tan88.2=O/A
31.82051595*5=
159.1025798
Height=159.1025798
Investigating a circle
First I have to use a equation to find out what the radius is once I found out the radius is I can find the area.
First I am going to take the equation 2πr
Then I take the equation 1000=2πr
From this I work out the radius: -
1000=2πr /2
500=πr /π (3.141592654)
159.1549431=r
Now I can work out the area of the circle using the equation πr²
πr²=π*159.1549431²= 79577.47155
Conclusion
The circle I found has the largest area compared with all the other shapes.
The area is larger because the more side you have to a shape the larger the area will become this is only when the perimeter equals the same on the shapes. I have worked out a formula to find out all the areas of shapes
Now I can draw a graph that will show how the areas increase as you get more sides.
I would suggest after these results that the farmer should make a circle out of the 1000 meters of fencing. This would give him the largest area for his cows to graze.
