area= 100 x 373 = 37300
a b
200m
As you can see from the results an equilateral triangle has the largest surface area of any triangle. This is interesting because an equilateral triangle has all equal sides. I believe that this will prove significant in later shapes as well.
I will now increase the number of sides by one to four, making them quadrilaterals.
This is a square and so has four equal sides (250m).
The surface area=250x250
A square has a surface area of 62500m2
I notice looking at this result that as well as having four equal sides it has a larger surface area than that of the largest triangle, (at this stage I do not think that this holds any significance to the investigation but I will note it any way; a square has14408m2 more than an equilateral triangle does).
This rectangle has is 450m in length and 50m in height.
Surface area=450x50
This rectangle has a surface area of 22500m2
Instead of drawing out all of the rectangles I shall just work out the areas
Looking at the results I can see that the square has the largest surface area out of all the rectangles.
An interesting pattern that I noticed in these results, is that, not only do the surface areas increase as the sides become more equal, but as the surface area increases it does so in ever smaller increments. The first increment is 17500 while the last is only 2500. The difference between the increments is constant, the surface area increases by 5000 less than the previous area did. As soon as the length goes beyond 250m the difference between surface area’s increase by 5000 each time. This suggests to me that a graph of these results would be similar to if not actually a parabola. To check this I will draw the graph
(it is stuck on the back of this sheet). The graph is indeed similar to a parabola (however I cannot prove if it is one or not) it also shows that the square does have the highest possible surface area out of the different rectangles.
Rectangles are only regular quadrilaterals therefore I will now investigate irregular quadrilaterals:
e
both sides marked e are 225m, side h is 300m and side w is 250m.
e h area = 225x225 + area of triangle involving line w
area of triangle = (300-225)x225 = 8437.5
------------- 2
w area= 50625+8437.5 = 59062.5
This is a kite. The top two sides are 200m and the bottom two sides are
300m. The angle at the top is 60o. It will require quite a lot of working to
find the area of this shape so I will do it on rough paper, (it will be
included with the rest of my work)
The area of this shape is 44700m2.
The areas of both of the irregular quadrilaterals are smaller than that of the square, therefore I can say that the square has the largest surface area of all quadrilaterals.
So far in both types shapes I have looked at, (triangle and quadrilateral), the shapes with all equal sides have had the highest surface area.
I think that the best way to go on from here is to explore different pentagons.
A regular pentagon with five equal sides has a surface area of 69400m2. First I divided it into five equal triangles then made those into right angled triangles. Then I used trigonometry to calculate the area of one of those. Then I multiplied that figure by five, (the working is on my rough paper).
I notice looking at this result that as well as having five equal sides the pentagon has a larger surface area than that of the largest rectangle, (the square), (at this stage I do not think that this holds any significance to the investigation but I will note it any way; a regular equally sided pentagon has 7200m2 more than the largest rectangle (square) does.
I think that regular pentagons will behave in the same way triangles and rectangles did. That is the pentagon with all sides even will have the largest surface area.
T T
This is an irregular pentagon. Sides marked H are 150m, sides marked
T are200m and side L is 300.
H H
L
Area of this shapes is: H x L + the area of the triangle on top.
150x300=45000.
To work out the area of the triangle I will use Pythagoras’s theorem. Height of the triangle = √(2002-1502) = 40000-22500 = √17500 = 132.2m
Area of the triangle = 132.2 x 150 = 19830m2
Area of the whole shape is 19830 + 45000 = 64830
Upon finding that a triangle, rectangle or pentagon that has all sides equal with a circumference of 1000m2 has a larger surface area than any other shape of the sa,e sort, I can reasonably assume that it will be the same with any other shape e.g. hexagons, heptagons and octagons. Therefore it follows that I will only have to measure the surface area of the regular shapes to discover which has the largest surface area.
At this point I have decided to work out my formulae (it took a long time and a LOT of scribbles so I have done it one rough paper).
Where: “n” is the number of sides.
“c” is the circumference (100m)
n c c / tan 360
2n 2n 2n m2
c
n c 2n
2n tan360
2n
Using this formulae I can find the area of any shape that has all sides equal length. Here are a few
As you can see as the number of sides on the shapes increase so does the area. This seems to suggest that the shape that has the most sides will have the highest area. The shape that has the most sides is a circle, because a circle has an infinite number of sides.
Using different formulas I can find the area of a circle with a circumference of 1000m.
Circumference is 1000m.
Diameter is 1000
╥
Area is ╥ D2
Area is 158845m2
Therefore the farmers 1000m of fence would be best put to use in fencing off a circle.
By
Jonathan Rusbridge
