Area =l x w, OR, l x b 1cm : 100m
Rectangle 1
Rectangle 2
Rectangle 3
More rectangles
Stage 2; Triangles
Now I plan to investigate other shapes to use to build the fence around the field so that I can find out if they give me a bigger area than the rectangles do.
The three triangles will have the same base and a perimeter of 1000m.
There are many types of triangles that I could use; scalene, right angled and isosceles, for example. Look at the triangles above, with a fixed base, I can see that they are all isosceles triangles which have the largest areas. Therefore I am only going to consider isosceles triangles for my task.
Method;
Draw three isosceles triangles with a total perimeter of 1000m each, and then calculate their areas.
Formula; Scale;
Area = s(s-a)(s-b)(s-c) 1cm : 100m
s=1/2 perimeter
Triangle 1
Triangle 2
Triangle 3
More triangles
A rectangular field has a bigger area because it has more sides.
Step 3; Zooming In
So I can gain a more accurate peak in my first Triangle graph I am planning to zoom in and find the areas of some of the triangle between 300m and 350m along the base.
Three equilateral triangles
1) Area = s(s-a)(s-b)(s-c)
= 1000(1000 -1000/3)(1000 -1000/3)(1000 -1000/3)
= 48 112.52m2
2) Area = s(s-a)(s-b)(s-c)
= 1000(1000 - 333)(1000 – 333)(1000 – 333)
= 48 112.45m2
3) Area = s(s-a)(s-b)(s-c)
= 1000(1000 – 334)(1000 - 334)(1000 – 334)
Looking at the graph we can see that the highest points are between 330m and 335m.
We know that out of all the rectangles a square had the greatest area. And as squares have all sides equal it made me wonder if that perhaps a triangle with equals sides and base would also have he greatest area out of all triangles.
After doing the sums above we know that an equilateral triangles has the biggest area out of them all.
Step 4; Polygons
Because my best rectangle was a square and my best triangle was an equilateral I have now decided that I should look at regular polygons as they also have sides that are of equal lengths.
I picked three polygons to draw, however there are several others that I could have used – seven sided, nine sided, etc.
Method;
Draw three polygons of a different number of sides with a total perimeter of 1000m each, and calculate their areas.
Formula; Scale;
b x h and Tan = Opposite 1cm : 100m
2 Adjacent
Polygon 1 – Pentagon
Polygon 2; Hexagon
Polygon 3; Octagon
More Polygons
As you can see by the table the area grows larger and larger as the polygons gain more sides. This means that the more sides the greater the area so we would want to build a field with as many sides as possible.
Looking at the diagrams you may notice that as the shapes gain more sides the lengths become shorter and they look more and more like a circle. Because of this I have decided to check out a circle and see how big an area that has.
Area = πr2
C = πd
1000 = πd
d = 1000 = 318.309
π
r = 1000 = 159.15
2π
Final Conclusion
We can now see that a circle has the greatest area of all shapes that we looked at, and therefore that is the best shape for the farmer to use when she builds her field.
